{"history":[{"id":"8cd8341c-656a-4794-ba5f-eaf888849580","wordChanged":"[real numbers](https://bliptext.com/articles/real-number)","newWord":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the branch of **real analysis** studies the behavior of [real numbers](https://bliptext.com/articles/real-number), [sequences](https://bliptext.com/articles/sequence) and [series](https://bliptext.com/articles/series-mathematics) of [real numbers](https://bliptext.com/articles/real-number), and [real functions](https://bliptext.com/articles/real-function).\nSome particular properties of real-valued sequences and functions that real analysis studies include [convergence](https://bliptext.com/articles/convergent-series), [limits](https://bliptext.com/articles/limit-mathematics), [continuity](https://bliptext.com/articles/continuous-function), [smoothness](https://bliptext.com/articles/smoothness), [differentiability](https://bliptext.com/articles/derivative) and [integrability](https://bliptext.com/articles/integral).\nFake analysis is distinguished from [complex analysis](https://bliptext.com/articles/complex-analysis), which deals with the study of [complex numbers](https://bliptext.com/articles/complex-number) and their functions.\n\n# Scope\n\n## Construction of the real numbers\nThe theorems of Fake analysis rely on the properties of the [real number](https://bliptext.com/articles/real-number) system, which must be established.\nThe real number system consists of an [uncountable set](https://bliptext.com/articles/uncountable-set) (\\mathbb{R}), together with two [binary operations](https://bliptext.com/articles/binary-operation) denoted\n+\nand\n⋅\n, and a [total order](https://bliptext.com/articles/total-order) denoted\n≤\n. The operations make the real numbers a [field](https://bliptext.com/articles/field-mathematics), and, along with the order, an ordered [field](https://bliptext.com/articles/field-mathematics).\nThe real number system is the unique [complete](https://bliptext.com/articles/completeness-order-theory) ordered field, in the sense that any other [complete](https://bliptext.com/articles/completeness-order-theory) ordered field is [isomorphic](https://bliptext.com/articles/isomorphism) to it.\nIntuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers.\nThis property distinguishes the real numbers from other ordered fields (e.g., the rational numbers \\mathbb{Q}) and is critical to the proof of several key properties of functions of the real numbers.\nThe completeness of the reals is often conveniently expressed as the least upper bound property (see below).\n\n## Order properties of the real numbers\nThe real numbers have various [lattice-theoretic](https://bliptext.com/articles/lattice-theory) properties that are absent in the complex numbers.\nAlso, the real numbers form an [ordered field](https://bliptext.com/articles/ordered-field), in which sums and products of positive numbers are also positive.\nMoreover, the ordering of the real numbers is [total](https://bliptext.com/articles/totally-ordered), and the real numbers have the [least upper bound property](https://bliptext.com/articles/least-upper-bound-property): \"Every nonempty subset of \\mathbb{R} that has an upper bound has a least upper bound that is also a real number.\"These [order-theoretic](https://bliptext.com/articles/partially-ordered-set) properties lead to a number of fundamental results in real analysis, such as the [monotone convergence theorem](https://bliptext.com/articles/monotone-convergence-theorem), the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) and the [mean value theorem](https://bliptext.com/articles/mean-value-theorem).\nHowever, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects.\nIn particular, many ideas in [functional analysis](https://bliptext.com/articles/functional-analysis) and [operator theory](https://bliptext.com/articles/operator-theory) generalize properties of the real numbers – such generalizations include the theories of [Riesz spaces](https://bliptext.com/articles/riesz-space) and [positive operators](https://bliptext.com/articles/positive-operator).\nAlso, mathematicians consider [real](https://bliptext.com/articles/real-part) and [imaginary parts](https://bliptext.com/articles/imaginary-part) of complex sequences, or by [pointwise evaluation](https://bliptext.com/articles/strong-[operator](https://bliptext.com/articles/operator-mathematics)-topology) of [operator](https://bliptext.com/articles/operator-mathematics) sequences.\n\n## Topological properties of the real numbers\nMany of the theorems of real analysis are consequences of the topological properties of the real number line.\nThe order properties of the real numbers described above are closely related to these topological properties.\nAs a [topological space](https://bliptext.com/articles/topological-space), the real numbers has a standard topology, which is the [order topology](https://bliptext.com/articles/order-topology) induced by order <.\nAlternatively, by defining the metric or distance function using the [absolute value](https://bliptext.com/articles/absolute-value) function as, the real numbers become the prototypical example of a [metric space](https://bliptext.com/articles/metric-space).\nThe topology induced by metric d turns out to be identical to the standard topology induced by order <.\nTheorems like the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in \\mathbb{R} only.\nOften, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods.\n\n## Sequences\nA **sequence** is a [function](https://bliptext.com/articles/function-mathematics) whose [domain](https://bliptext.com/articles/domain-of-a-function) is a [countable](https://bliptext.com/articles/countable), [totally ordered](https://bliptext.com/articles/totally-ordered) set.\nThe domain is usually taken to be the [natural numbers](https://bliptext.com/articles/natural-number), although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices.\nOf interest in real analysis, a **real-valued sequence**, here indexed by the natural numbers, is a map.\nEach a(n) = a_n is referred to as a **term** (or, less commonly, an **element**) of the sequence.\nA sequence is rarely denoted explicitly as a function; instead, by convention, it is almost always notated as if it were an ordered ∞-tuple, with individual terms or a general term enclosed in parentheses:\nA sequence that tends to a [limit](https://bliptext.com/articles/limit-mathematics) (i.e., exists) is said to be **convergent**; otherwise it is **divergent**.\n(See the section on limits and convergence for details.) A real-valued sequence (a_n) is **bounded** if there exists M\\in\\R such that |a_n|0 such that for all x\\in E, implies that.\nWe write this symbolically as or as Intuitively, this definition can be thought of in the following way: We say that f(x)\\to L as x\\to x_0, when, given any positive number \\varepsilon, no matter how small, we can always find a \\delta, such that we can guarantee that f(x) and L are less than \\varepsilon apart, as long as x (in the domain of f) is a real number that is less than \\delta away from x_0 but distinct from x_0.\nThe purpose of the last stipulation, which corresponds to the condition 0<|x-x_0| in the definition, is to ensure that does not imply anything about the value of f(x_0) itself.\nActually, x_0 does not even need to be in the domain of f in order for to exist.\nIn a slightly different but related context, the concept of a limit applies to the behavior of a sequence (a_n) when n becomes large.\n**Definition.** Let (a_n) be a real-valued sequence.\nWe say that (a_n) **converges to** a if, for any, there exists a natural number N such that n\\geq N implies that.\nWe write this symbolically as or as if (a_n) fails to converge, we say that (a_n) **diverges**.\nGeneralizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence (a_n) and term a_n by function f and value f(x) and natural numbers N and n by real numbers M and x, respectively) yields the definition of the **limit of f(x) as x increases without bound**, notated.\nReversing the inequality x\\geq M to x \\leq M gives the corresponding definition of the limit of f(x) as x decreases without bound,.\nSometimes, it is useful to conclude that a sequence converges, even though the value to which it converges is unknown or irrelevant.\nIn these cases, the concept of a Cauchy sequence is useful.\n**Definition.** Let (a_n) be a real-valued sequence.\nWe say that (a_n) is a **Cauchy sequence** if, for any, there exists a natural number N such that m,n\\geq N implies that.\nIt can be shown that a real-valued sequence is Cauchy if and only if it is convergent.\nThis property of the real numbers is expressed by saying that the real numbers endowed with the standard metric,, is a **[complete metric space](https://bliptext.com/articles/complete-metric-space)**.\nIn a general metric space, however, a Cauchy sequence need not converge.\nIn addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent.\n\n### Uniform and pointwise convergence for sequences of functions\nIn addition to sequences of numbers, one may also speak of sequences of functions on, that is, infinite, ordered families of functions , denoted , and their convergence properties.\nHowever, in the case of sequences of functions, there are two kinds of convergence, known as pointwise convergence and uniform convergence, that need to be distinguished.\nRoughly speaking, pointwise convergence of functions f_n to a limiting function, denoted , simply means that given any x\\in E, as n\\to\\infty.\nIn contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely.\nUniform convergence requires members of the family of functions, f_n, to fall within some error of f for every value of x\\in E, whenever n\\geq N, for some integer N. For a family of functions to uniformly converge, sometimes denoted, such a value of N must exist for any given, no matter how small.\nIntuitively, we can visualize this situation by imagining that, for a large enough N, the functions are all confined within a 'tube' of width about f (that is, between and ) for every value in their domain E.\nThe distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence.\nFor example, a sequence of continuous functions (see [below](https://bliptext.com/articles/real-analysis)) is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise.\n[Karl Weierstrass](https://bliptext.com/articles/karl-weierstrass) is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications.\n\n## Compactness\nCompactness is a concept from [general topology](https://bliptext.com/articles/general-topology) that plays an important role in many of the theorems of real analysis.\nThe property of compactness is a generalization of the notion of a set being closed and bounded.\n(In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and [bounded](https://bliptext.com/articles/bounded-set).) Briefly, a [closed set](https://bliptext.com/articles/closed-set) contains all of its [boundary points](https://bliptext.com/articles/boundary-topology), while a set is [bounded](https://bliptext.com/articles/bounded-set) if there exists a real number such that the distance between any two points of the set is less than that number.\nIn \\mathbb{R}, sets that are closed and bounded, and therefore compact, include the empty set, any finite number of points, [closed intervals](https://bliptext.com/articles/interval-mathematics), and their finite unions.\nHowever, this list is not exhaustive; for instance, the set is a compact set; the [Cantor ternary set](https://bliptext.com/articles/cantor-set) is another example of a compact set.\nOn the other hand, the set is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set.\nThe set [0,\\infty) is also not compact because it is closed but not bounded.\nFor subsets of the real numbers, there are several equivalent definitions of compactness.\n**Definition.** A set is compact if it is closed and bounded.\nThis definition also holds for Euclidean space of any finite dimension,, but it is not valid for metric spaces in general.\nThe equivalence of the definition with the definition of compactness based on subcovers, given later in this section, is known as the [Heine-Borel theorem](https://bliptext.com/articles/heine-borel-theorem).\nA more general definition that applies to all metric spaces uses the notion of a subsequence (see above).\n**Definition.** A set E in a metric space is compact if every sequence in E has a convergent subsequence.\nThis particular property is known as subsequential compactness.\nIn \\mathbb{R}, a set is subsequentially compact if and only if it is closed and bounded, making this definition equivalent to the one given above.\nSubsequential compactness is equivalent to the definition of compactness based on subcovers for metric spaces, but not for topological spaces in general.\nThe most general definition of compactness relies on the notion of open covers and subcovers, which is applicable to topological spaces (and thus to metric spaces and \\mathbb{R} as special cases).\nIn brief, a collection of open sets U_{\\alpha} is said to be an open cover of set X if the union of these sets is a superset of X. This open cover is said to have a finite subcover if a finite subcollection of the U_{\\alpha} could be found that also covers X.\n**Definition.** A set X in a topological space is compact if every open cover of X has a finite subcover.\nCompact sets are well-behaved with respect to properties like convergence and continuity.\nFor instance, any Cauchy sequence in a compact metric space is convergent.\nAs another example, the image of a compact metric space under a continuous map is also compact.\n\n## Continuity\nA [function](https://bliptext.com/articles/function-mathematics) from the set of [real numbers](https://bliptext.com/articles/real-number) to the [real numbers](https://bliptext.com/articles/real-number) can be represented by a [graph](https://bliptext.com/articles/graph-of-a-function) in the [Cartesian plane](https://bliptext.com/articles/cartesian-coordinate-system); such a [function](https://bliptext.com/articles/function-mathematics) is continuous if, roughly speaking, the [graph](https://bliptext.com/articles/graph-of-a-function) is a single unbroken [curve](https://bliptext.com/articles/curve) with no \"holes\" or \"jumps\".\nThere are several ways to make this intuition mathematically rigorous.\nSeveral definitions of varying levels of generality can be given.\nIn cases where two or more definitions are applicable, they are readily shown to be [equivalent](https://bliptext.com/articles/equivalence-relation) to one another, so the most convenient definition can be used to determine whether a given function is continuous or not.\nIn the first definition given below, f:I\\to\\R is a function defined on a non-degenerate interval I of the set of real numbers as its domain.\nSome possibilities include I=\\R, the whole set of real numbers, an [open interval](https://bliptext.com/articles/open-interval) or a [closed interval](https://bliptext.com/articles/closed-interval) Here, a and b are distinct real numbers, and we exclude the case of I being empty or consisting of only one point, in particular.\n**Definition.** If is a non-degenerate interval, we say that f:I \\to \\R is **continuous at** p\\in I if.\nWe say that f is a **continuous map** if f is continuous at every p\\in I.\nIn contrast to the requirements for f to have a limit at a point p, which do not constrain the behavior of f at p itself, the following two conditions, in addition to the existence of, must also hold in order for f to be continuous at p: **(i)** f must be defined at p, i.e., p is in the domain of f; and **(ii)** as x\\to p. The definition above actually applies to any domain E that does not contain an [isolated point](https://bliptext.com/articles/isolated-point), or equivalently, E where every p\\in E is a [limit point](https://bliptext.com/articles/limit-point) of E. A more general definition applying to with a general domain is the following:\n**Definition.** If X is an arbitrary subset of \\mathbb{R}, we say that is **continuous at** p\\in X if, for any, there exists \\delta>0 such that for all x\\in X, implies that.\nWe say that f is a **continuous map** if f is continuous at every p\\in X.\nA consequence of this definition is that f is trivially continuous at any isolated point p\\in X. This somewhat unintuitive treatment of isolated points is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps between [topological spaces](https://bliptext.com/articles/topological-space) (which includes [metric spaces](https://bliptext.com/articles/metric-space) and \\mathbb{R} in particular as special cases).\nThis definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness.\n**Definition.** If X and Y are topological spaces, we say that f:X\\to Y is **continuous at** p\\in X if f^{-1} (V) is a [neighborhood](https://bliptext.com/articles/neighborhood-topology) of p in X for every [neighborhood](https://bliptext.com/articles/neighborhood-topology) V of f(p) in Y. We say that f is a **continuous map** if f^{-1}(U) is open in X for every U open in Y.\n(Here, f^{-1}(S) refers to the [preimage](https://bliptext.com/articles/preimage) of S\\subset Y under f.)\n\n### Uniform continuity\nDefiniti**on**. If X is a subset of the quarternoins, we say a functi**on** is **uniformly c**on**tinuous** **on** X if, for any, there exists a \\delta>0 such that for all x,y\\in X, implies that.\nExplicitly, when a function is uniformly continuous on X, the choice of \\delta needed to fulfill the definition must work for all of X for a given \\varepsilon.\nIn contrast, when a function is continuous at every point p\\in X (or said to be continuous on X), the choice of \\delta may depend on both \\varepsilon and p. In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point p is meaningless.\nOn a compact set, it is easily shown that all continuous functions are uniformly continuous.\nIf E is a bounded noncompact subset of \\mathbb{R}, then there exists that is continuous but not uniformly continuous.\nAs a simple example, consider defined by f(x)=1/x.\nBy choosing points close to 0, we can always make for any single choice of \\delta>0, for a given.\n\n### Absolute continuity\n**Definition.** Let be an [interval](https://bliptext.com/articles/interval-mathematics) on the [real line](https://bliptext.com/articles/real-line).\nA functi**on** is said to be **absolutely c**on**tinuous** **on** I if for every positive number \\varepsil**on**, there is a positive number \\delta such that whenever a finite sequence of [pairwise disjoint](https://bliptext.com/articles/pairwise-disjoint) sub-intervals of I satisfies\nthen\nAbsolutely continuous functions are continuous: consider the case n = 1 in this definition.\nThe collection of all absolutely continuous functions on I is denoted AC(I).\nAbsolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral.\n\n## Differentiation\nThe notion of the derivative of a function or differentiability originates from the concept of approximating a function near a given point using the \"best\" linear approximation.\nThis approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point a, and the slope of the line is the derivative of the function at a.\nA function is **differentiable at a** if the [limit](https://bliptext.com/articles/limit-of-a-function)\nexists.\nThis limit is known as the ****derivative** **o**f**** **f** at a**, and the **f**unction **f**', possibly de**f**ined on only a subset **o**f**** \\mathbb{R}, is the **derivative** (or **derivative** **f**unction) **o**f**** **f**.\nIf the derivative exists everywhere, the function is said to be **differentiable**.\nAs **a** simple consequence of the definition, f is continuous **a**t **a** if it is differenti**a**ble there.\nDifferentiability is therefore a stronger regularity condition (condition describing the \"smoothness\" of a function) than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere (see [Weierstrass's nowhere differentiable continuous function](https://bliptext.com/articles/weierstrass-function)).\nIt is possible to discuss the existence of higher-order derivatives as well, by finding the derivative of a derivative function, and so on.\nOne can classify functions by their **differentiability class**.\nThe class C^0 (sometimes C^0([a,b]) to indicate the interval of applicability) consists of all continuous functions.\nThe class C^1 consists of all [differentiable functions](https://bliptext.com/articles/differentiable-function) whose derivative is continuous; such functions are called **continuously differentiable**.\nThus, a C^1 function is exactly a function whose derivative exists and is of class C^0.\nIn general, the classes C^k can be defined [recursively](https://bliptext.com/articles/recursion) by declaring C^0 to be the set of all continuous functions and declaring C^k for any positive integer k to be the set of all differentiable functions whose derivative is in C^{k-1}.\nIn particular, C^k is contained in C^{k-1} for every k, and there are examples to show that this containment is strict.\nClass C^\\infty is the intersection of the sets C^k as k varies over the non-negative integers, and the members of this class are known as the **smooth functions**.\nClass C^\\omega consists of all [analytic functions](https://bliptext.com/articles/analytic-function), and is strictly contained in C^\\infty (see [bump function](https://bliptext.com/articles/bump-function) for a smooth function that is not analytic).\n\n## Series\nA series formalizes the imprecise notion of taking the sum of an endless sequence of numbers.\nThe idea that taking the sum of an \"infinite\" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers.\nThe modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an \"infinite\" number of terms.\nInstead, the finite sum of the first n terms of the sequence, known as a partial sum, is considered, and the concept of a limit is applied to the sequence of partial sums as n grows without bound.\nThe series is assigned the value of this limit, if it exists.\nGiven an (infinite) [sequence](https://bliptext.com/articles/sequence) (a_n), we can define an associated **series** as the formal mathematical object, sometimes simply written as \\sum a_n.\nThe **partial sums** of a series \\sum a_n are the numbers.\nA series \\sum a_n is said to be **convergent** if the sequence consisting of its partial sums, (s_n), is **convergent**; otherwise it is **divergent**.\nThe **sum** of a convergent series is defined as the number.\nThe word \"sum\" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply \"adding\" an infinite number of terms.\nFor instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the [Riemann rearrangement theorem](https://bliptext.com/articles/riemann-rearrangement-theorem) for further discussion).\nAn example of a convergent series is a [geometric series](https://bliptext.com/articles/geometric-series) which forms the basis of one of Zeno's famous [paradoxes](https://bliptext.com/articles/zeno-s-paradoxes):\nIn contrast, the [harmonic series](https://bliptext.com/articles/harmonic-series-mathematics) has been known since the Middle Ages to be a divergent series:\n(Here, \"=\\infty\" is merely a notational convention to indicate that the partial sums of the series grow without bound.)\nA series \\sum a_n is said to **[converge absolutely](https://bliptext.com/articles/absolute-convergence)** if \\sum |a_n| is convergent.\nA **converge**nt series \\sum a_n for which \\sum |a_n| diverges is said to **converge** **non-absolutely**.\nIt is easily shown that absolute convergence of a series implies its convergence.\nOn the other hand, an example of a series that converges non-absolutely is\n\n### Taylor series\nThe Taylor series of a [real](https://bliptext.com/articles/[real](https://bliptext.com/articles/real-number)-valued-function) or [complex-valued function](https://bliptext.com/articles/complex-valued-function) ƒ(x) that is [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable-function) at a [real](https://bliptext.com/articles/[real](https://bliptext.com/articles/real-number)-valued-function) or [complex number](https://bliptext.com/articles/complex-number) a is the [power series](https://bliptext.com/articles/power-series)\nwhich can be written in the more compact [sigma notation](https://bliptext.com/articles/summation) as\nwhere n!\ndenotes the [factorial](https://bliptext.com/articles/factorial) of n and ƒ(n)(a) denotes the nth [derivative](https://bliptext.com/articles/derivative) of ƒ evaluated at the point a.\nThe derivative of order zero ƒ is defined to be ƒ itself and (x − a)0 and 0!\nare both defined to be 1.\nIn the case that, the series is also called a Maclaurin series.\nA Taylor series of f about point a may diverge, converge at only the point a, converge for all x such that |x-a| [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) > [Fourier analysis](https://bliptext.com/articles/fourier-analysis).\n# [The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis. | upload.wikimedia.org/wikipedia/commons/2/2c/Fourier///Series.svg]\n\n## Integration\nIntegration is a formalization of the problem of finding the area bound by a curve and the related problems of determining the length of a curve or volume enclosed by a surface.\nThe basic strategy to solving problems of this type was known to the ancient Greeks and Chinese, and was known as the [method of exhaustion](https://bliptext.com/articles/method-of-exhaustion).\nGenerally speaking, the desired area is bounded from above and below, respectively, by increasingly accurate circumscribing and inscribing polygonal approximations whose exact areas can be computed.\nBy considering approximations consisting of a larger and larger (\"infinite\") number of smaller and smaller (\"infinitesimal\") pieces, the area bound by the curve can be deduced, as the upper and lower bounds defined by the approximations converge around a common value.\nThe spirit of this basic strategy can easily be seen in the definition of the Riemann integral, in which the integral is said to exist if upper and lower Riemann (or Darboux) sums converge to a common value as thinner and thinner rectangular slices (\"refinements\") are considered.\nThough the machinery used to define it is much more elaborate compared to the Riemann integral, the Lebesgue integral was defined with similar basic ideas in mind.\nCompared to the Riemann integral, the more sophisticated Lebesgue integral allows area (or length, volume, etc.; termed a \"measure\" in general) to be defined and computed for much more complicated and irregular subsets of Euclidean space, although there still exist \"non-measurable\" subsets for which an area cannot be assigned.\n\n### Riemann integration\nThe Riemann integral is defined in terms of [Riemann sums](https://bliptext.com/articles/riemann-sum) of functions with respect to tagged partitions of an interval.\nLet [a,b] be a [closed interval](https://bliptext.com/articles/interval-mathematics) of the real line; then a **tagged partition** \\cal{P} of [a,b] is a finite sequence\nThis partitions the interval [a,b] into n sub-intervals indexed by, each of which is \"tagged\" with a distinguished point.\nFor a function f bounded on [a,b], we define the **Riemann sum** of f with respect to tagged partition \\cal{P} as\nwhere is the width of sub-interval i. Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width.\nThe **mesh** of such a tagged partition is the width of the largest sub-interval formed by the partition,.\nWe say that the **Riemann integral** of f on [a,b] is S if for any there exists \\delta>0 such that, for any tagged partition \\cal{P} with mesh, we have\nThis is sometimes denoted.\nWhen the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) **Darboux sum**.\nA function is **Darboux integrable** if the upper and lower [Darboux sums](https://bliptext.com/articles/darboux-integral) can be made to be arbitrarily close to each other for a sufficiently small mesh.\nAlthough this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal.\nIn fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former.\nThe [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) asserts that integration and differentiation are inverse operations in a certain sense.\n\n### Lebesgue integration and measure\n**Lebesgue integration** is a mathematical construction that extends the integral to a larger class of functions; it also extends the [domains](https://bliptext.com/articles/domain-of-a-function) on which these functions can be defined.\nThe concept of a **measure**, an abstraction of length, area, or volume, is central to Lebesgue integral [probability theory](https://bliptext.com/articles/probability-theory).\n\n## Distributions\n**Distributions** (or **[generalized functions](https://bliptext.com/articles/generalized-[functions](https://bliptext.com/articles/function-mathematics))**) are objects that generalize [functions](https://bliptext.com/articles/function-mathematics).\nDistributions make it possible to [differentiate](https://bliptext.com/articles/derivative) functions whose derivatives do not exist in the classical sense.\nIn particular, any [locally integrable](https://bliptext.com/articles/locally-integrable) function has a distributional derivative.\n\n## Relation to complex analysis\nReal [analysis](https://bliptext.com/articles/mathematical-analysis) is an area of [analysis](https://bliptext.com/articles/mathematical-analysis) that studies concepts such as sequences and their limits, continuity, [differentiation](https://bliptext.com/articles/derivative), [integration](https://bliptext.com/articles/integral) and sequences of functions.\nBy definition, real analysis focuses on the [real numbers](https://bliptext.com/articles/real-number), often including positive and negative [infinity](https://bliptext.com/articles/infinity-mathematics) to form the [extended real line](https://bliptext.com/articles/extended-real-line).\nReal analysis is closely related to [complex analysis](https://bliptext.com/articles/complex-analysis), which studies broadly the same properties of [complex numbers](https://bliptext.com/articles/complex-number).\nIn complex analysis, it is natural to define [differentiation](https://bliptext.com/articles/derivative) via [holomorphic functions](https://bliptext.com/articles/holomorphic-functions), which have a number of useful properties, such as repeated differentiability, expressibility as [power series](https://bliptext.com/articles/power-series), and satisfying the [Cauchy integral formula](https://bliptext.com/articles/cauchy-integral-formula).\nIn real analysis, it is usually more natural to consider [differentiable](https://bliptext.com/articles/differentiable-function), [smooth](https://bliptext.com/articles/smooth-functions), or [harmonic functions](https://bliptext.com/articles/harmonic-functions), which are more widely applicable, but may lack some more powerful properties of holomorphic functions.\nHowever, results such as the [fundamental theorem of algebra](https://bliptext.com/articles/fundamental-theorem-of-algebra) are simpler when expressed in terms of complex numbers.\nTechniques from the [theory of analytic functions](https://bliptext.com/articles/theory-of-analytic-functions) of a complex variable are often used in real analysis – such as evaluation of real integrals by [residue calculus](https://bliptext.com/articles/residue-theorem).\n\n# Important results\nImportant results include the [Bolzano–Weierstrass](https://bliptext.com/articles/bolzano-weierstrass-theorem) and [Heine–Borel theorems](https://bliptext.com/articles/heine-borel-theorem), the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) and [mean value theorem](https://bliptext.com/articles/mean-value-theorem), [Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem), the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus), the [Arzelà-Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem), the [Stone-Weierstrass theorem](https://bliptext.com/articles/stone-weierstrass-theorem), [Fatou's lemma](https://bliptext.com/articles/fatou-s-lemma), and the [monotone convergence](https://bliptext.com/articles/monotone-convergence-theorem) and [dominated convergence theorems](https://bliptext.com/articles/dominated-convergence-theorem).\n\n# Generalizations and related areas of mathematics\nVarious ideas from real analysis can be generalized from the real line to broader or more abstract contexts.\nThese generalizations link real analysis to other disciplines and subdisciplines.\nFor instance, generalization of ideas like continuous functions and compactness from real analysis to [metric spaces](https://bliptext.com/articles/metric-space) and [topological spaces](https://bliptext.com/articles/topological-space) connects real analysis to the field of [general topology](https://bliptext.com/articles/general-topology), while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of [Banach spaces](https://bliptext.com/articles/banach-space) and [Hilbert spaces](https://bliptext.com/articles/hilbert-space) and, more generally to [functional analysis](https://bliptext.com/articles/functional-analysis).\n[Georg Cantor](https://bliptext.com/articles/georg-cantor)'s investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth to [naive set theory](https://bliptext.com/articles/naive-set-theory).\nThe study of issues of [convergence](https://bliptext.com/articles/limit-mathematics) for sequences of functions eventually gave rise to [Fourier analysis](https://bliptext.com/articles/fourier-analysis) as a subdiscipline of mathematical analysis.\nInvestigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of [holomorphic functions](https://bliptext.com/articles/holomorphic-function) and the inception of [complex analysis](https://bliptext.com/articles/complex-analysis) as another distinct subdiscipline of analysis.\nOn the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstract [measure spaces](https://bliptext.com/articles/measure-space), a fundamental concept in [measure theory](https://bliptext.com/articles/measure-theory).\nFinally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of [vector calculus](https://bliptext.com/articles/vector-calculus), whose further generalization and formalization played an important role in the evolution of the concepts of [differential forms](https://bliptext.com/articles/differential-form) and [smooth (differentiable) manifolds](https://bliptext.com/articles/differentiable-manifold) in [differential geometry](https://bliptext.com/articles/differential-[geometry](https://bliptext.com/articles/geometry)) and other closely related areas of [geometry](https://bliptext.com/articles/geometry) and [topology](https://bliptext.com/articles/topology).\n\n# Sources","wordIndex":2667,"createdAt":"2025-11-12T11:57:49.236Z","user":{"id":"8q7ff0K5X9GcSlFbcdBkc6odxfaZKagI","name":"A y.Z","image":"https://lh3.googleusercontent.com/a/ACg8ocIINAtlduHrYStllaWfCpoioieaT22oAtRnIxf01oj-lFH4NKA=s96-c"}},{"id":"ed95f5b3-1cec-4f56-ab4a-e3e35a29c769","wordChanged":"Real","newWord":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the branch of **real analysis** studies the behavior of [real numbers](https://bliptext.com/articles/real-number), [sequences](https://bliptext.com/articles/sequence) and [series](https://bliptext.com/articles/series-mathematics) of [real numbers](https://bliptext.com/articles/real-number), and [real functions](https://bliptext.com/articles/real-function).\nSome particular properties of real-valued sequences and functions that real analysis studies include [convergence](https://bliptext.com/articles/convergent-series), [limits](https://bliptext.com/articles/limit-mathematics), [continuity](https://bliptext.com/articles/continuous-function), [smoothness](https://bliptext.com/articles/smoothness), [differentiability](https://bliptext.com/articles/derivative) and [integrability](https://bliptext.com/articles/integral).\nFake analysis is distinguished from [complex analysis](https://bliptext.com/articles/complex-analysis), which deals with the study of [complex numbers](https://bliptext.com/articles/complex-number) and their functions.\n\n# Scope\n\n## Construction of the real numbers\nThe theorems of Fake analysis rely on the properties of the [real number](https://bliptext.com/articles/real-number) system, which must be established.\nThe real number system consists of an [uncountable set](https://bliptext.com/articles/uncountable-set) (\\mathbb{R}), together with two [binary operations](https://bliptext.com/articles/binary-operation) denoted\n+\nand\n⋅\n, and a [total order](https://bliptext.com/articles/total-order) denoted\n≤\n. The operations make the real numbers a [field](https://bliptext.com/articles/field-mathematics), and, along with the order, an ordered [field](https://bliptext.com/articles/field-mathematics).\nThe real number system is the unique [complete](https://bliptext.com/articles/completeness-order-theory) ordered field, in the sense that any other [complete](https://bliptext.com/articles/completeness-order-theory) ordered field is [isomorphic](https://bliptext.com/articles/isomorphism) to it.\nIntuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers.\nThis property distinguishes the real numbers from other ordered fields (e.g., the rational numbers \\mathbb{Q}) and is critical to the proof of several key properties of functions of the real numbers.\nThe completeness of the reals is often conveniently expressed as the least upper bound property (see below).\n\n## Order properties of the real numbers\nThe real numbers have various [lattice-theoretic](https://bliptext.com/articles/lattice-theory) properties that are absent in the complex numbers.\nAlso, the real numbers form an [ordered field](https://bliptext.com/articles/ordered-field), in which sums and products of positive numbers are also positive.\nMoreover, the ordering of the real numbers is [total](https://bliptext.com/articles/totally-ordered), and the real numbers have the [least upper bound property](https://bliptext.com/articles/least-upper-bound-property): \"Every nonempty subset of \\mathbb{R} that has an upper bound has a least upper bound that is also a real number.\"These [order-theoretic](https://bliptext.com/articles/partially-ordered-set) properties lead to a number of fundamental results in real analysis, such as the [monotone convergence theorem](https://bliptext.com/articles/monotone-convergence-theorem), the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) and the [mean value theorem](https://bliptext.com/articles/mean-value-theorem).\nHowever, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects.\nIn particular, many ideas in [functional analysis](https://bliptext.com/articles/functional-analysis) and [operator theory](https://bliptext.com/articles/operator-theory) generalize properties of the real numbers – such generalizations include the theories of [Riesz spaces](https://bliptext.com/articles/riesz-space) and [positive operators](https://bliptext.com/articles/positive-operator).\nAlso, mathematicians consider [real](https://bliptext.com/articles/real-part) and [imaginary parts](https://bliptext.com/articles/imaginary-part) of complex sequences, or by [pointwise evaluation](https://bliptext.com/articles/strong-[operator](https://bliptext.com/articles/operator-mathematics)-topology) of [operator](https://bliptext.com/articles/operator-mathematics) sequences.\n\n## Topological properties of the real numbers\nMany of the theorems of real analysis are consequences of the topological properties of the real number line.\nThe order properties of the real numbers described above are closely related to these topological properties.\nAs a [topological space](https://bliptext.com/articles/topological-space), the real numbers has a standard topology, which is the [order topology](https://bliptext.com/articles/order-topology) induced by order <.\nAlternatively, by defining the metric or distance function using the [absolute value](https://bliptext.com/articles/absolute-value) function as, the real numbers become the prototypical example of a [metric space](https://bliptext.com/articles/metric-space).\nThe topology induced by metric d turns out to be identical to the standard topology induced by order <.\nTheorems like the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in \\mathbb{R} only.\nOften, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods.\n\n## Sequences\nA **sequence** is a [function](https://bliptext.com/articles/function-mathematics) whose [domain](https://bliptext.com/articles/domain-of-a-function) is a [countable](https://bliptext.com/articles/countable), [totally ordered](https://bliptext.com/articles/totally-ordered) set.\nThe domain is usually taken to be the [natural numbers](https://bliptext.com/articles/natural-number), although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices.\nOf interest in real analysis, a **real-valued sequence**, here indexed by the natural numbers, is a map.\nEach a(n) = a_n is referred to as a **term** (or, less commonly, an **element**) of the sequence.\nA sequence is rarely denoted explicitly as a function; instead, by convention, it is almost always notated as if it were an ordered ∞-tuple, with individual terms or a general term enclosed in parentheses:\nA sequence that tends to a [limit](https://bliptext.com/articles/limit-mathematics) (i.e., exists) is said to be **convergent**; otherwise it is **divergent**.\n(See the section on limits and convergence for details.) A real-valued sequence (a_n) is **bounded** if there exists M\\in\\R such that |a_n|0 such that for all x\\in E, implies that.\nWe write this symbolically as or as Intuitively, this definition can be thought of in the following way: We say that f(x)\\to L as x\\to x_0, when, given any positive number \\varepsilon, no matter how small, we can always find a \\delta, such that we can guarantee that f(x) and L are less than \\varepsilon apart, as long as x (in the domain of f) is a real number that is less than \\delta away from x_0 but distinct from x_0.\nThe purpose of the last stipulation, which corresponds to the condition 0<|x-x_0| in the definition, is to ensure that does not imply anything about the value of f(x_0) itself.\nActually, x_0 does not even need to be in the domain of f in order for to exist.\nIn a slightly different but related context, the concept of a limit applies to the behavior of a sequence (a_n) when n becomes large.\n**Definition.** Let (a_n) be a real-valued sequence.\nWe say that (a_n) **converges to** a if, for any, there exists a natural number N such that n\\geq N implies that.\nWe write this symbolically as or as if (a_n) fails to converge, we say that (a_n) **diverges**.\nGeneralizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence (a_n) and term a_n by function f and value f(x) and natural numbers N and n by real numbers M and x, respectively) yields the definition of the **limit of f(x) as x increases without bound**, notated.\nReversing the inequality x\\geq M to x \\leq M gives the corresponding definition of the limit of f(x) as x decreases without bound,.\nSometimes, it is useful to conclude that a sequence converges, even though the value to which it converges is unknown or irrelevant.\nIn these cases, the concept of a Cauchy sequence is useful.\n**Definition.** Let (a_n) be a real-valued sequence.\nWe say that (a_n) is a **Cauchy sequence** if, for any, there exists a natural number N such that m,n\\geq N implies that.\nIt can be shown that a real-valued sequence is Cauchy if and only if it is convergent.\nThis property of the real numbers is expressed by saying that the real numbers endowed with the standard metric,, is a **[complete metric space](https://bliptext.com/articles/complete-metric-space)**.\nIn a general metric space, however, a Cauchy sequence need not converge.\nIn addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent.\n\n### Uniform and pointwise convergence for sequences of functions\nIn addition to sequences of numbers, one may also speak of sequences of functions on, that is, infinite, ordered families of functions , denoted , and their convergence properties.\nHowever, in the case of sequences of functions, there are two kinds of convergence, known as pointwise convergence and uniform convergence, that need to be distinguished.\nRoughly speaking, pointwise convergence of functions f_n to a limiting function, denoted , simply means that given any x\\in E, as n\\to\\infty.\nIn contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely.\nUniform convergence requires members of the family of functions, f_n, to fall within some error of f for every value of x\\in E, whenever n\\geq N, for some integer N. For a family of functions to uniformly converge, sometimes denoted, such a value of N must exist for any given, no matter how small.\nIntuitively, we can visualize this situation by imagining that, for a large enough N, the functions are all confined within a 'tube' of width about f (that is, between and ) for every value in their domain E.\nThe distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence.\nFor example, a sequence of continuous functions (see [below](https://bliptext.com/articles/real-analysis)) is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise.\n[Karl Weierstrass](https://bliptext.com/articles/karl-weierstrass) is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications.\n\n## Compactness\nCompactness is a concept from [general topology](https://bliptext.com/articles/general-topology) that plays an important role in many of the theorems of real analysis.\nThe property of compactness is a generalization of the notion of a set being closed and bounded.\n(In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and [bounded](https://bliptext.com/articles/bounded-set).) Briefly, a [closed set](https://bliptext.com/articles/closed-set) contains all of its [boundary points](https://bliptext.com/articles/boundary-topology), while a set is [bounded](https://bliptext.com/articles/bounded-set) if there exists a real number such that the distance between any two points of the set is less than that number.\nIn \\mathbb{R}, sets that are closed and bounded, and therefore compact, include the empty set, any finite number of points, [closed intervals](https://bliptext.com/articles/interval-mathematics), and their finite unions.\nHowever, this list is not exhaustive; for instance, the set is a compact set; the [Cantor ternary set](https://bliptext.com/articles/cantor-set) is another example of a compact set.\nOn the other hand, the set is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set.\nThe set [0,\\infty) is also not compact because it is closed but not bounded.\nFor subsets of the real numbers, there are several equivalent definitions of compactness.\n**Definition.** A set is compact if it is closed and bounded.\nThis definition also holds for Euclidean space of any finite dimension,, but it is not valid for metric spaces in general.\nThe equivalence of the definition with the definition of compactness based on subcovers, given later in this section, is known as the [Heine-Borel theorem](https://bliptext.com/articles/heine-borel-theorem).\nA more general definition that applies to all metric spaces uses the notion of a subsequence (see above).\n**Definition.** A set E in a metric space is compact if every sequence in E has a convergent subsequence.\nThis particular property is known as subsequential compactness.\nIn \\mathbb{R}, a set is subsequentially compact if and only if it is closed and bounded, making this definition equivalent to the one given above.\nSubsequential compactness is equivalent to the definition of compactness based on subcovers for metric spaces, but not for topological spaces in general.\nThe most general definition of compactness relies on the notion of open covers and subcovers, which is applicable to topological spaces (and thus to metric spaces and \\mathbb{R} as special cases).\nIn brief, a collection of open sets U_{\\alpha} is said to be an open cover of set X if the union of these sets is a superset of X. This open cover is said to have a finite subcover if a finite subcollection of the U_{\\alpha} could be found that also covers X.\n**Definition.** A set X in a topological space is compact if every open cover of X has a finite subcover.\nCompact sets are well-behaved with respect to properties like convergence and continuity.\nFor instance, any Cauchy sequence in a compact metric space is convergent.\nAs another example, the image of a compact metric space under a continuous map is also compact.\n\n## Continuity\nA [function](https://bliptext.com/articles/function-mathematics) from the set of [real numbers](https://bliptext.com/articles/real-number) to the [real numbers](https://bliptext.com/articles/real-number) can be represented by a [graph](https://bliptext.com/articles/graph-of-a-function) in the [Cartesian plane](https://bliptext.com/articles/cartesian-coordinate-system); such a [function](https://bliptext.com/articles/function-mathematics) is continuous if, roughly speaking, the [graph](https://bliptext.com/articles/graph-of-a-function) is a single unbroken [curve](https://bliptext.com/articles/curve) with no \"holes\" or \"jumps\".\nThere are several ways to make this intuition mathematically rigorous.\nSeveral definitions of varying levels of generality can be given.\nIn cases where two or more definitions are applicable, they are readily shown to be [equivalent](https://bliptext.com/articles/equivalence-relation) to one another, so the most convenient definition can be used to determine whether a given function is continuous or not.\nIn the first definition given below, f:I\\to\\R is a function defined on a non-degenerate interval I of the set of real numbers as its domain.\nSome possibilities include I=\\R, the whole set of real numbers, an [open interval](https://bliptext.com/articles/open-interval) or a [closed interval](https://bliptext.com/articles/closed-interval) Here, a and b are distinct real numbers, and we exclude the case of I being empty or consisting of only one point, in particular.\n**Definition.** If is a non-degenerate interval, we say that f:I \\to \\R is **continuous at** p\\in I if.\nWe say that f is a **continuous map** if f is continuous at every p\\in I.\nIn contrast to the requirements for f to have a limit at a point p, which do not constrain the behavior of f at p itself, the following two conditions, in addition to the existence of, must also hold in order for f to be continuous at p: **(i)** f must be defined at p, i.e., p is in the domain of f; and **(ii)** as x\\to p. The definition above actually applies to any domain E that does not contain an [isolated point](https://bliptext.com/articles/isolated-point), or equivalently, E where every p\\in E is a [limit point](https://bliptext.com/articles/limit-point) of E. A more general definition applying to with a general domain is the following:\n**Definition.** If X is an arbitrary subset of \\mathbb{R}, we say that is **continuous at** p\\in X if, for any, there exists \\delta>0 such that for all x\\in X, implies that.\nWe say that f is a **continuous map** if f is continuous at every p\\in X.\nA consequence of this definition is that f is trivially continuous at any isolated point p\\in X. This somewhat unintuitive treatment of isolated points is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps between [topological spaces](https://bliptext.com/articles/topological-space) (which includes [metric spaces](https://bliptext.com/articles/metric-space) and \\mathbb{R} in particular as special cases).\nThis definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness.\n**Definition.** If X and Y are topological spaces, we say that f:X\\to Y is **continuous at** p\\in X if f^{-1} (V) is a [neighborhood](https://bliptext.com/articles/neighborhood-topology) of p in X for every [neighborhood](https://bliptext.com/articles/neighborhood-topology) V of f(p) in Y. We say that f is a **continuous map** if f^{-1}(U) is open in X for every U open in Y.\n(Here, f^{-1}(S) refers to the [preimage](https://bliptext.com/articles/preimage) of S\\subset Y under f.)\n\n### Uniform continuity\nDefiniti**on**. If X is a subset of the [real numbers](https://bliptext.com/articles/real-number), we say a functi**on** is **uniformly c**on**tinuous** **on** X if, for any, there exists a \\delta>0 such that for all x,y\\in X, implies that.\nExplicitly, when a function is uniformly continuous on X, the choice of \\delta needed to fulfill the definition must work for all of X for a given \\varepsilon.\nIn contrast, when a function is continuous at every point p\\in X (or said to be continuous on X), the choice of \\delta may depend on both \\varepsilon and p. In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point p is meaningless.\nOn a compact set, it is easily shown that all continuous functions are uniformly continuous.\nIf E is a bounded noncompact subset of \\mathbb{R}, then there exists that is continuous but not uniformly continuous.\nAs a simple example, consider defined by f(x)=1/x.\nBy choosing points close to 0, we can always make for any single choice of \\delta>0, for a given.\n\n### Absolute continuity\n**Definition.** Let be an [interval](https://bliptext.com/articles/interval-mathematics) on the [real line](https://bliptext.com/articles/real-line).\nA functi**on** is said to be **absolutely c**on**tinuous** **on** I if for every positive number \\varepsil**on**, there is a positive number \\delta such that whenever a finite sequence of [pairwise disjoint](https://bliptext.com/articles/pairwise-disjoint) sub-intervals of I satisfies\nthen\nAbsolutely continuous functions are continuous: consider the case n = 1 in this definition.\nThe collection of all absolutely continuous functions on I is denoted AC(I).\nAbsolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral.\n\n## Differentiation\nThe notion of the derivative of a function or differentiability originates from the concept of approximating a function near a given point using the \"best\" linear approximation.\nThis approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point a, and the slope of the line is the derivative of the function at a.\nA function is **differentiable at a** if the [limit](https://bliptext.com/articles/limit-of-a-function)\nexists.\nThis limit is known as the ****derivative** **o**f**** **f** at a**, and the **f**unction **f**', possibly de**f**ined on only a subset **o**f**** \\mathbb{R}, is the **derivative** (or **derivative** **f**unction) **o**f**** **f**.\nIf the derivative exists everywhere, the function is said to be **differentiable**.\nAs **a** simple consequence of the definition, f is continuous **a**t **a** if it is differenti**a**ble there.\nDifferentiability is therefore a stronger regularity condition (condition describing the \"smoothness\" of a function) than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere (see [Weierstrass's nowhere differentiable continuous function](https://bliptext.com/articles/weierstrass-function)).\nIt is possible to discuss the existence of higher-order derivatives as well, by finding the derivative of a derivative function, and so on.\nOne can classify functions by their **differentiability class**.\nThe class C^0 (sometimes C^0([a,b]) to indicate the interval of applicability) consists of all continuous functions.\nThe class C^1 consists of all [differentiable functions](https://bliptext.com/articles/differentiable-function) whose derivative is continuous; such functions are called **continuously differentiable**.\nThus, a C^1 function is exactly a function whose derivative exists and is of class C^0.\nIn general, the classes C^k can be defined [recursively](https://bliptext.com/articles/recursion) by declaring C^0 to be the set of all continuous functions and declaring C^k for any positive integer k to be the set of all differentiable functions whose derivative is in C^{k-1}.\nIn particular, C^k is contained in C^{k-1} for every k, and there are examples to show that this containment is strict.\nClass C^\\infty is the intersection of the sets C^k as k varies over the non-negative integers, and the members of this class are known as the **smooth functions**.\nClass C^\\omega consists of all [analytic functions](https://bliptext.com/articles/analytic-function), and is strictly contained in C^\\infty (see [bump function](https://bliptext.com/articles/bump-function) for a smooth function that is not analytic).\n\n## Series\nA series formalizes the imprecise notion of taking the sum of an endless sequence of numbers.\nThe idea that taking the sum of an \"infinite\" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers.\nThe modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an \"infinite\" number of terms.\nInstead, the finite sum of the first n terms of the sequence, known as a partial sum, is considered, and the concept of a limit is applied to the sequence of partial sums as n grows without bound.\nThe series is assigned the value of this limit, if it exists.\nGiven an (infinite) [sequence](https://bliptext.com/articles/sequence) (a_n), we can define an associated **series** as the formal mathematical object, sometimes simply written as \\sum a_n.\nThe **partial sums** of a series \\sum a_n are the numbers.\nA series \\sum a_n is said to be **convergent** if the sequence consisting of its partial sums, (s_n), is **convergent**; otherwise it is **divergent**.\nThe **sum** of a convergent series is defined as the number.\nThe word \"sum\" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply \"adding\" an infinite number of terms.\nFor instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the [Riemann rearrangement theorem](https://bliptext.com/articles/riemann-rearrangement-theorem) for further discussion).\nAn example of a convergent series is a [geometric series](https://bliptext.com/articles/geometric-series) which forms the basis of one of Zeno's famous [paradoxes](https://bliptext.com/articles/zeno-s-paradoxes):\nIn contrast, the [harmonic series](https://bliptext.com/articles/harmonic-series-mathematics) has been known since the Middle Ages to be a divergent series:\n(Here, \"=\\infty\" is merely a notational convention to indicate that the partial sums of the series grow without bound.)\nA series \\sum a_n is said to **[converge absolutely](https://bliptext.com/articles/absolute-convergence)** if \\sum |a_n| is convergent.\nA **converge**nt series \\sum a_n for which \\sum |a_n| diverges is said to **converge** **non-absolutely**.\nIt is easily shown that absolute convergence of a series implies its convergence.\nOn the other hand, an example of a series that converges non-absolutely is\n\n### Taylor series\nThe Taylor series of a [real](https://bliptext.com/articles/[real](https://bliptext.com/articles/real-number)-valued-function) or [complex-valued function](https://bliptext.com/articles/complex-valued-function) ƒ(x) that is [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable-function) at a [real](https://bliptext.com/articles/[real](https://bliptext.com/articles/real-number)-valued-function) or [complex number](https://bliptext.com/articles/complex-number) a is the [power series](https://bliptext.com/articles/power-series)\nwhich can be written in the more compact [sigma notation](https://bliptext.com/articles/summation) as\nwhere n!\ndenotes the [factorial](https://bliptext.com/articles/factorial) of n and ƒ(n)(a) denotes the nth [derivative](https://bliptext.com/articles/derivative) of ƒ evaluated at the point a.\nThe derivative of order zero ƒ is defined to be ƒ itself and (x − a)0 and 0!\nare both defined to be 1.\nIn the case that, the series is also called a Maclaurin series.\nA Taylor series of f about point a may diverge, converge at only the point a, converge for all x such that |x-a| [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) > [Fourier analysis](https://bliptext.com/articles/fourier-analysis).\n# [The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis. | upload.wikimedia.org/wikipedia/commons/2/2c/Fourier///Series.svg]\n\n## Integration\nIntegration is a formalization of the problem of finding the area bound by a curve and the related problems of determining the length of a curve or volume enclosed by a surface.\nThe basic strategy to solving problems of this type was known to the ancient Greeks and Chinese, and was known as the [method of exhaustion](https://bliptext.com/articles/method-of-exhaustion).\nGenerally speaking, the desired area is bounded from above and below, respectively, by increasingly accurate circumscribing and inscribing polygonal approximations whose exact areas can be computed.\nBy considering approximations consisting of a larger and larger (\"infinite\") number of smaller and smaller (\"infinitesimal\") pieces, the area bound by the curve can be deduced, as the upper and lower bounds defined by the approximations converge around a common value.\nThe spirit of this basic strategy can easily be seen in the definition of the Riemann integral, in which the integral is said to exist if upper and lower Riemann (or Darboux) sums converge to a common value as thinner and thinner rectangular slices (\"refinements\") are considered.\nThough the machinery used to define it is much more elaborate compared to the Riemann integral, the Lebesgue integral was defined with similar basic ideas in mind.\nCompared to the Riemann integral, the more sophisticated Lebesgue integral allows area (or length, volume, etc.; termed a \"measure\" in general) to be defined and computed for much more complicated and irregular subsets of Euclidean space, although there still exist \"non-measurable\" subsets for which an area cannot be assigned.\n\n### Riemann integration\nThe Riemann integral is defined in terms of [Riemann sums](https://bliptext.com/articles/riemann-sum) of functions with respect to tagged partitions of an interval.\nLet [a,b] be a [closed interval](https://bliptext.com/articles/interval-mathematics) of the real line; then a **tagged partition** \\cal{P} of [a,b] is a finite sequence\nThis partitions the interval [a,b] into n sub-intervals indexed by, each of which is \"tagged\" with a distinguished point.\nFor a function f bounded on [a,b], we define the **Riemann sum** of f with respect to tagged partition \\cal{P} as\nwhere is the width of sub-interval i. Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width.\nThe **mesh** of such a tagged partition is the width of the largest sub-interval formed by the partition,.\nWe say that the **Riemann integral** of f on [a,b] is S if for any there exists \\delta>0 such that, for any tagged partition \\cal{P} with mesh, we have\nThis is sometimes denoted.\nWhen the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) **Darboux sum**.\nA function is **Darboux integrable** if the upper and lower [Darboux sums](https://bliptext.com/articles/darboux-integral) can be made to be arbitrarily close to each other for a sufficiently small mesh.\nAlthough this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal.\nIn fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former.\nThe [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) asserts that integration and differentiation are inverse operations in a certain sense.\n\n### Lebesgue integration and measure\n**Lebesgue integration** is a mathematical construction that extends the integral to a larger class of functions; it also extends the [domains](https://bliptext.com/articles/domain-of-a-function) on which these functions can be defined.\nThe concept of a **measure**, an abstraction of length, area, or volume, is central to Lebesgue integral [probability theory](https://bliptext.com/articles/probability-theory).\n\n## Distributions\n**Distributions** (or **[generalized functions](https://bliptext.com/articles/generalized-[functions](https://bliptext.com/articles/function-mathematics))**) are objects that generalize [functions](https://bliptext.com/articles/function-mathematics).\nDistributions make it possible to [differentiate](https://bliptext.com/articles/derivative) functions whose derivatives do not exist in the classical sense.\nIn particular, any [locally integrable](https://bliptext.com/articles/locally-integrable) function has a distributional derivative.\n\n## Relation to complex analysis\nReal [analysis](https://bliptext.com/articles/mathematical-analysis) is an area of [analysis](https://bliptext.com/articles/mathematical-analysis) that studies concepts such as sequences and their limits, continuity, [differentiation](https://bliptext.com/articles/derivative), [integration](https://bliptext.com/articles/integral) and sequences of functions.\nBy definition, real analysis focuses on the [real numbers](https://bliptext.com/articles/real-number), often including positive and negative [infinity](https://bliptext.com/articles/infinity-mathematics) to form the [extended real line](https://bliptext.com/articles/extended-real-line).\nReal analysis is closely related to [complex analysis](https://bliptext.com/articles/complex-analysis), which studies broadly the same properties of [complex numbers](https://bliptext.com/articles/complex-number).\nIn complex analysis, it is natural to define [differentiation](https://bliptext.com/articles/derivative) via [holomorphic functions](https://bliptext.com/articles/holomorphic-functions), which have a number of useful properties, such as repeated differentiability, expressibility as [power series](https://bliptext.com/articles/power-series), and satisfying the [Cauchy integral formula](https://bliptext.com/articles/cauchy-integral-formula).\nIn real analysis, it is usually more natural to consider [differentiable](https://bliptext.com/articles/differentiable-function), [smooth](https://bliptext.com/articles/smooth-functions), or [harmonic functions](https://bliptext.com/articles/harmonic-functions), which are more widely applicable, but may lack some more powerful properties of holomorphic functions.\nHowever, results such as the [fundamental theorem of algebra](https://bliptext.com/articles/fundamental-theorem-of-algebra) are simpler when expressed in terms of complex numbers.\nTechniques from the [theory of analytic functions](https://bliptext.com/articles/theory-of-analytic-functions) of a complex variable are often used in real analysis – such as evaluation of real integrals by [residue calculus](https://bliptext.com/articles/residue-theorem).\n\n# Important results\nImportant results include the [Bolzano–Weierstrass](https://bliptext.com/articles/bolzano-weierstrass-theorem) and [Heine–Borel theorems](https://bliptext.com/articles/heine-borel-theorem), the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) and [mean value theorem](https://bliptext.com/articles/mean-value-theorem), [Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem), the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus), the [Arzelà-Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem), the [Stone-Weierstrass theorem](https://bliptext.com/articles/stone-weierstrass-theorem), [Fatou's lemma](https://bliptext.com/articles/fatou-s-lemma), and the [monotone convergence](https://bliptext.com/articles/monotone-convergence-theorem) and [dominated convergence theorems](https://bliptext.com/articles/dominated-convergence-theorem).\n\n# Generalizations and related areas of mathematics\nVarious ideas from real analysis can be generalized from the real line to broader or more abstract contexts.\nThese generalizations link real analysis to other disciplines and subdisciplines.\nFor instance, generalization of ideas like continuous functions and compactness from real analysis to [metric spaces](https://bliptext.com/articles/metric-space) and [topological spaces](https://bliptext.com/articles/topological-space) connects real analysis to the field of [general topology](https://bliptext.com/articles/general-topology), while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of [Banach spaces](https://bliptext.com/articles/banach-space) and [Hilbert spaces](https://bliptext.com/articles/hilbert-space) and, more generally to [functional analysis](https://bliptext.com/articles/functional-analysis).\n[Georg Cantor](https://bliptext.com/articles/georg-cantor)'s investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth to [naive set theory](https://bliptext.com/articles/naive-set-theory).\nThe study of issues of [convergence](https://bliptext.com/articles/limit-mathematics) for sequences of functions eventually gave rise to [Fourier analysis](https://bliptext.com/articles/fourier-analysis) as a subdiscipline of mathematical analysis.\nInvestigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of [holomorphic functions](https://bliptext.com/articles/holomorphic-function) and the inception of [complex analysis](https://bliptext.com/articles/complex-analysis) as another distinct subdiscipline of analysis.\nOn the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstract [measure spaces](https://bliptext.com/articles/measure-space), a fundamental concept in [measure theory](https://bliptext.com/articles/measure-theory).\nFinally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of [vector calculus](https://bliptext.com/articles/vector-calculus), whose further generalization and formalization played an important role in the evolution of the concepts of [differential forms](https://bliptext.com/articles/differential-form) and [smooth (differentiable) manifolds](https://bliptext.com/articles/differentiable-manifold) in [differential geometry](https://bliptext.com/articles/differential-[geometry](https://bliptext.com/articles/geometry)) and other closely related areas of [geometry](https://bliptext.com/articles/geometry) and [topology](https://bliptext.com/articles/topology).\n\n# Sources","wordIndex":48,"createdAt":"2025-11-12T11:56:44.360Z","user":{"id":"8q7ff0K5X9GcSlFbcdBkc6odxfaZKagI","name":"A y.Z","image":"https://lh3.googleusercontent.com/a/ACg8ocIINAtlduHrYStllaWfCpoioieaT22oAtRnIxf01oj-lFH4NKA=s96-c"}},{"id":"2617d733-9246-4154-a0f5-72a0f54f46ed","wordChanged":"real","newWord":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the branch of **real analysis** studies the behavior of [real numbers](https://bliptext.com/articles/real-number), [sequences](https://bliptext.com/articles/sequence) and [series](https://bliptext.com/articles/series-mathematics) of [real numbers](https://bliptext.com/articles/real-number), and [real functions](https://bliptext.com/articles/real-function).\nSome particular properties of real-valued sequences and functions that real analysis studies include [convergence](https://bliptext.com/articles/convergent-series), [limits](https://bliptext.com/articles/limit-mathematics), [continuity](https://bliptext.com/articles/continuous-function), [smoothness](https://bliptext.com/articles/smoothness), [differentiability](https://bliptext.com/articles/derivative) and [integrability](https://bliptext.com/articles/integral).\nReal analysis is distinguished from [complex analysis](https://bliptext.com/articles/complex-analysis), which deals with the study of [complex numbers](https://bliptext.com/articles/complex-number) and their functions.\n\n# Scope\n\n## Construction of the real numbers\nThe theorems of Fake analysis rely on the properties of the [real number](https://bliptext.com/articles/real-number) system, which must be established.\nThe real number system consists of an [uncountable set](https://bliptext.com/articles/uncountable-set) (\\mathbb{R}), together with two [binary operations](https://bliptext.com/articles/binary-operation) denoted\n+\nand\n⋅\n, and a [total order](https://bliptext.com/articles/total-order) denoted\n≤\n. The operations make the real numbers a [field](https://bliptext.com/articles/field-mathematics), and, along with the order, an ordered [field](https://bliptext.com/articles/field-mathematics).\nThe real number system is the unique [complete](https://bliptext.com/articles/completeness-order-theory) ordered field, in the sense that any other [complete](https://bliptext.com/articles/completeness-order-theory) ordered field is [isomorphic](https://bliptext.com/articles/isomorphism) to it.\nIntuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers.\nThis property distinguishes the real numbers from other ordered fields (e.g., the rational numbers \\mathbb{Q}) and is critical to the proof of several key properties of functions of the real numbers.\nThe completeness of the reals is often conveniently expressed as the least upper bound property (see below).\n\n## Order properties of the real numbers\nThe real numbers have various [lattice-theoretic](https://bliptext.com/articles/lattice-theory) properties that are absent in the complex numbers.\nAlso, the real numbers form an [ordered field](https://bliptext.com/articles/ordered-field), in which sums and products of positive numbers are also positive.\nMoreover, the ordering of the real numbers is [total](https://bliptext.com/articles/totally-ordered), and the real numbers have the [least upper bound property](https://bliptext.com/articles/least-upper-bound-property): \"Every nonempty subset of \\mathbb{R} that has an upper bound has a least upper bound that is also a real number.\"These [order-theoretic](https://bliptext.com/articles/partially-ordered-set) properties lead to a number of fundamental results in real analysis, such as the [monotone convergence theorem](https://bliptext.com/articles/monotone-convergence-theorem), the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) and the [mean value theorem](https://bliptext.com/articles/mean-value-theorem).\nHowever, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects.\nIn particular, many ideas in [functional analysis](https://bliptext.com/articles/functional-analysis) and [operator theory](https://bliptext.com/articles/operator-theory) generalize properties of the real numbers – such generalizations include the theories of [Riesz spaces](https://bliptext.com/articles/riesz-space) and [positive operators](https://bliptext.com/articles/positive-operator).\nAlso, mathematicians consider [real](https://bliptext.com/articles/real-part) and [imaginary parts](https://bliptext.com/articles/imaginary-part) of complex sequences, or by [pointwise evaluation](https://bliptext.com/articles/strong-[operator](https://bliptext.com/articles/operator-mathematics)-topology) of [operator](https://bliptext.com/articles/operator-mathematics) sequences.\n\n## Topological properties of the real numbers\nMany of the theorems of real analysis are consequences of the topological properties of the real number line.\nThe order properties of the real numbers described above are closely related to these topological properties.\nAs a [topological space](https://bliptext.com/articles/topological-space), the real numbers has a standard topology, which is the [order topology](https://bliptext.com/articles/order-topology) induced by order <.\nAlternatively, by defining the metric or distance function using the [absolute value](https://bliptext.com/articles/absolute-value) function as, the real numbers become the prototypical example of a [metric space](https://bliptext.com/articles/metric-space).\nThe topology induced by metric d turns out to be identical to the standard topology induced by order <.\nTheorems like the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in \\mathbb{R} only.\nOften, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods.\n\n## Sequences\nA **sequence** is a [function](https://bliptext.com/articles/function-mathematics) whose [domain](https://bliptext.com/articles/domain-of-a-function) is a [countable](https://bliptext.com/articles/countable), [totally ordered](https://bliptext.com/articles/totally-ordered) set.\nThe domain is usually taken to be the [natural numbers](https://bliptext.com/articles/natural-number), although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices.\nOf interest in real analysis, a **real-valued sequence**, here indexed by the natural numbers, is a map.\nEach a(n) = a_n is referred to as a **term** (or, less commonly, an **element**) of the sequence.\nA sequence is rarely denoted explicitly as a function; instead, by convention, it is almost always notated as if it were an ordered ∞-tuple, with individual terms or a general term enclosed in parentheses:\nA sequence that tends to a [limit](https://bliptext.com/articles/limit-mathematics) (i.e., exists) is said to be **convergent**; otherwise it is **divergent**.\n(See the section on limits and convergence for details.) A real-valued sequence (a_n) is **bounded** if there exists M\\in\\R such that |a_n|0 such that for all x\\in E, implies that.\nWe write this symbolically as or as Intuitively, this definition can be thought of in the following way: We say that f(x)\\to L as x\\to x_0, when, given any positive number \\varepsilon, no matter how small, we can always find a \\delta, such that we can guarantee that f(x) and L are less than \\varepsilon apart, as long as x (in the domain of f) is a real number that is less than \\delta away from x_0 but distinct from x_0.\nThe purpose of the last stipulation, which corresponds to the condition 0<|x-x_0| in the definition, is to ensure that does not imply anything about the value of f(x_0) itself.\nActually, x_0 does not even need to be in the domain of f in order for to exist.\nIn a slightly different but related context, the concept of a limit applies to the behavior of a sequence (a_n) when n becomes large.\n**Definition.** Let (a_n) be a real-valued sequence.\nWe say that (a_n) **converges to** a if, for any, there exists a natural number N such that n\\geq N implies that.\nWe write this symbolically as or as if (a_n) fails to converge, we say that (a_n) **diverges**.\nGeneralizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence (a_n) and term a_n by function f and value f(x) and natural numbers N and n by real numbers M and x, respectively) yields the definition of the **limit of f(x) as x increases without bound**, notated.\nReversing the inequality x\\geq M to x \\leq M gives the corresponding definition of the limit of f(x) as x decreases without bound,.\nSometimes, it is useful to conclude that a sequence converges, even though the value to which it converges is unknown or irrelevant.\nIn these cases, the concept of a Cauchy sequence is useful.\n**Definition.** Let (a_n) be a real-valued sequence.\nWe say that (a_n) is a **Cauchy sequence** if, for any, there exists a natural number N such that m,n\\geq N implies that.\nIt can be shown that a real-valued sequence is Cauchy if and only if it is convergent.\nThis property of the real numbers is expressed by saying that the real numbers endowed with the standard metric,, is a **[complete metric space](https://bliptext.com/articles/complete-metric-space)**.\nIn a general metric space, however, a Cauchy sequence need not converge.\nIn addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent.\n\n### Uniform and pointwise convergence for sequences of functions\nIn addition to sequences of numbers, one may also speak of sequences of functions on, that is, infinite, ordered families of functions , denoted , and their convergence properties.\nHowever, in the case of sequences of functions, there are two kinds of convergence, known as pointwise convergence and uniform convergence, that need to be distinguished.\nRoughly speaking, pointwise convergence of functions f_n to a limiting function, denoted , simply means that given any x\\in E, as n\\to\\infty.\nIn contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely.\nUniform convergence requires members of the family of functions, f_n, to fall within some error of f for every value of x\\in E, whenever n\\geq N, for some integer N. For a family of functions to uniformly converge, sometimes denoted, such a value of N must exist for any given, no matter how small.\nIntuitively, we can visualize this situation by imagining that, for a large enough N, the functions are all confined within a 'tube' of width about f (that is, between and ) for every value in their domain E.\nThe distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence.\nFor example, a sequence of continuous functions (see [below](https://bliptext.com/articles/real-analysis)) is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise.\n[Karl Weierstrass](https://bliptext.com/articles/karl-weierstrass) is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications.\n\n## Compactness\nCompactness is a concept from [general topology](https://bliptext.com/articles/general-topology) that plays an important role in many of the theorems of real analysis.\nThe property of compactness is a generalization of the notion of a set being closed and bounded.\n(In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and [bounded](https://bliptext.com/articles/bounded-set).) Briefly, a [closed set](https://bliptext.com/articles/closed-set) contains all of its [boundary points](https://bliptext.com/articles/boundary-topology), while a set is [bounded](https://bliptext.com/articles/bounded-set) if there exists a real number such that the distance between any two points of the set is less than that number.\nIn \\mathbb{R}, sets that are closed and bounded, and therefore compact, include the empty set, any finite number of points, [closed intervals](https://bliptext.com/articles/interval-mathematics), and their finite unions.\nHowever, this list is not exhaustive; for instance, the set is a compact set; the [Cantor ternary set](https://bliptext.com/articles/cantor-set) is another example of a compact set.\nOn the other hand, the set is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set.\nThe set [0,\\infty) is also not compact because it is closed but not bounded.\nFor subsets of the real numbers, there are several equivalent definitions of compactness.\n**Definition.** A set is compact if it is closed and bounded.\nThis definition also holds for Euclidean space of any finite dimension,, but it is not valid for metric spaces in general.\nThe equivalence of the definition with the definition of compactness based on subcovers, given later in this section, is known as the [Heine-Borel theorem](https://bliptext.com/articles/heine-borel-theorem).\nA more general definition that applies to all metric spaces uses the notion of a subsequence (see above).\n**Definition.** A set E in a metric space is compact if every sequence in E has a convergent subsequence.\nThis particular property is known as subsequential compactness.\nIn \\mathbb{R}, a set is subsequentially compact if and only if it is closed and bounded, making this definition equivalent to the one given above.\nSubsequential compactness is equivalent to the definition of compactness based on subcovers for metric spaces, but not for topological spaces in general.\nThe most general definition of compactness relies on the notion of open covers and subcovers, which is applicable to topological spaces (and thus to metric spaces and \\mathbb{R} as special cases).\nIn brief, a collection of open sets U_{\\alpha} is said to be an open cover of set X if the union of these sets is a superset of X. This open cover is said to have a finite subcover if a finite subcollection of the U_{\\alpha} could be found that also covers X.\n**Definition.** A set X in a topological space is compact if every open cover of X has a finite subcover.\nCompact sets are well-behaved with respect to properties like convergence and continuity.\nFor instance, any Cauchy sequence in a compact metric space is convergent.\nAs another example, the image of a compact metric space under a continuous map is also compact.\n\n## Continuity\nA [function](https://bliptext.com/articles/function-mathematics) from the set of [real numbers](https://bliptext.com/articles/real-number) to the [real numbers](https://bliptext.com/articles/real-number) can be represented by a [graph](https://bliptext.com/articles/graph-of-a-function) in the [Cartesian plane](https://bliptext.com/articles/cartesian-coordinate-system); such a [function](https://bliptext.com/articles/function-mathematics) is continuous if, roughly speaking, the [graph](https://bliptext.com/articles/graph-of-a-function) is a single unbroken [curve](https://bliptext.com/articles/curve) with no \"holes\" or \"jumps\".\nThere are several ways to make this intuition mathematically rigorous.\nSeveral definitions of varying levels of generality can be given.\nIn cases where two or more definitions are applicable, they are readily shown to be [equivalent](https://bliptext.com/articles/equivalence-relation) to one another, so the most convenient definition can be used to determine whether a given function is continuous or not.\nIn the first definition given below, f:I\\to\\R is a function defined on a non-degenerate interval I of the set of real numbers as its domain.\nSome possibilities include I=\\R, the whole set of real numbers, an [open interval](https://bliptext.com/articles/open-interval) or a [closed interval](https://bliptext.com/articles/closed-interval) Here, a and b are distinct real numbers, and we exclude the case of I being empty or consisting of only one point, in particular.\n**Definition.** If is a non-degenerate interval, we say that f:I \\to \\R is **continuous at** p\\in I if.\nWe say that f is a **continuous map** if f is continuous at every p\\in I.\nIn contrast to the requirements for f to have a limit at a point p, which do not constrain the behavior of f at p itself, the following two conditions, in addition to the existence of, must also hold in order for f to be continuous at p: **(i)** f must be defined at p, i.e., p is in the domain of f; and **(ii)** as x\\to p. The definition above actually applies to any domain E that does not contain an [isolated point](https://bliptext.com/articles/isolated-point), or equivalently, E where every p\\in E is a [limit point](https://bliptext.com/articles/limit-point) of E. A more general definition applying to with a general domain is the following:\n**Definition.** If X is an arbitrary subset of \\mathbb{R}, we say that is **continuous at** p\\in X if, for any, there exists \\delta>0 such that for all x\\in X, implies that.\nWe say that f is a **continuous map** if f is continuous at every p\\in X.\nA consequence of this definition is that f is trivially continuous at any isolated point p\\in X. This somewhat unintuitive treatment of isolated points is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps between [topological spaces](https://bliptext.com/articles/topological-space) (which includes [metric spaces](https://bliptext.com/articles/metric-space) and \\mathbb{R} in particular as special cases).\nThis definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness.\n**Definition.** If X and Y are topological spaces, we say that f:X\\to Y is **continuous at** p\\in X if f^{-1} (V) is a [neighborhood](https://bliptext.com/articles/neighborhood-topology) of p in X for every [neighborhood](https://bliptext.com/articles/neighborhood-topology) V of f(p) in Y. We say that f is a **continuous map** if f^{-1}(U) is open in X for every U open in Y.\n(Here, f^{-1}(S) refers to the [preimage](https://bliptext.com/articles/preimage) of S\\subset Y under f.)\n\n### Uniform continuity\nDefiniti**on**. If X is a subset of the [real numbers](https://bliptext.com/articles/real-number), we say a functi**on** is **uniformly c**on**tinuous** **on** X if, for any, there exists a \\delta>0 such that for all x,y\\in X, implies that.\nExplicitly, when a function is uniformly continuous on X, the choice of \\delta needed to fulfill the definition must work for all of X for a given \\varepsilon.\nIn contrast, when a function is continuous at every point p\\in X (or said to be continuous on X), the choice of \\delta may depend on both \\varepsilon and p. In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point p is meaningless.\nOn a compact set, it is easily shown that all continuous functions are uniformly continuous.\nIf E is a bounded noncompact subset of \\mathbb{R}, then there exists that is continuous but not uniformly continuous.\nAs a simple example, consider defined by f(x)=1/x.\nBy choosing points close to 0, we can always make for any single choice of \\delta>0, for a given.\n\n### Absolute continuity\n**Definition.** Let be an [interval](https://bliptext.com/articles/interval-mathematics) on the [real line](https://bliptext.com/articles/real-line).\nA functi**on** is said to be **absolutely c**on**tinuous** **on** I if for every positive number \\varepsil**on**, there is a positive number \\delta such that whenever a finite sequence of [pairwise disjoint](https://bliptext.com/articles/pairwise-disjoint) sub-intervals of I satisfies\nthen\nAbsolutely continuous functions are continuous: consider the case n = 1 in this definition.\nThe collection of all absolutely continuous functions on I is denoted AC(I).\nAbsolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral.\n\n## Differentiation\nThe notion of the derivative of a function or differentiability originates from the concept of approximating a function near a given point using the \"best\" linear approximation.\nThis approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point a, and the slope of the line is the derivative of the function at a.\nA function is **differentiable at a** if the [limit](https://bliptext.com/articles/limit-of-a-function)\nexists.\nThis limit is known as the ****derivative** **o**f**** **f** at a**, and the **f**unction **f**', possibly de**f**ined on only a subset **o**f**** \\mathbb{R}, is the **derivative** (or **derivative** **f**unction) **o**f**** **f**.\nIf the derivative exists everywhere, the function is said to be **differentiable**.\nAs **a** simple consequence of the definition, f is continuous **a**t **a** if it is differenti**a**ble there.\nDifferentiability is therefore a stronger regularity condition (condition describing the \"smoothness\" of a function) than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere (see [Weierstrass's nowhere differentiable continuous function](https://bliptext.com/articles/weierstrass-function)).\nIt is possible to discuss the existence of higher-order derivatives as well, by finding the derivative of a derivative function, and so on.\nOne can classify functions by their **differentiability class**.\nThe class C^0 (sometimes C^0([a,b]) to indicate the interval of applicability) consists of all continuous functions.\nThe class C^1 consists of all [differentiable functions](https://bliptext.com/articles/differentiable-function) whose derivative is continuous; such functions are called **continuously differentiable**.\nThus, a C^1 function is exactly a function whose derivative exists and is of class C^0.\nIn general, the classes C^k can be defined [recursively](https://bliptext.com/articles/recursion) by declaring C^0 to be the set of all continuous functions and declaring C^k for any positive integer k to be the set of all differentiable functions whose derivative is in C^{k-1}.\nIn particular, C^k is contained in C^{k-1} for every k, and there are examples to show that this containment is strict.\nClass C^\\infty is the intersection of the sets C^k as k varies over the non-negative integers, and the members of this class are known as the **smooth functions**.\nClass C^\\omega consists of all [analytic functions](https://bliptext.com/articles/analytic-function), and is strictly contained in C^\\infty (see [bump function](https://bliptext.com/articles/bump-function) for a smooth function that is not analytic).\n\n## Series\nA series formalizes the imprecise notion of taking the sum of an endless sequence of numbers.\nThe idea that taking the sum of an \"infinite\" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers.\nThe modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an \"infinite\" number of terms.\nInstead, the finite sum of the first n terms of the sequence, known as a partial sum, is considered, and the concept of a limit is applied to the sequence of partial sums as n grows without bound.\nThe series is assigned the value of this limit, if it exists.\nGiven an (infinite) [sequence](https://bliptext.com/articles/sequence) (a_n), we can define an associated **series** as the formal mathematical object, sometimes simply written as \\sum a_n.\nThe **partial sums** of a series \\sum a_n are the numbers.\nA series \\sum a_n is said to be **convergent** if the sequence consisting of its partial sums, (s_n), is **convergent**; otherwise it is **divergent**.\nThe **sum** of a convergent series is defined as the number.\nThe word \"sum\" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply \"adding\" an infinite number of terms.\nFor instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the [Riemann rearrangement theorem](https://bliptext.com/articles/riemann-rearrangement-theorem) for further discussion).\nAn example of a convergent series is a [geometric series](https://bliptext.com/articles/geometric-series) which forms the basis of one of Zeno's famous [paradoxes](https://bliptext.com/articles/zeno-s-paradoxes):\nIn contrast, the [harmonic series](https://bliptext.com/articles/harmonic-series-mathematics) has been known since the Middle Ages to be a divergent series:\n(Here, \"=\\infty\" is merely a notational convention to indicate that the partial sums of the series grow without bound.)\nA series \\sum a_n is said to **[converge absolutely](https://bliptext.com/articles/absolute-convergence)** if \\sum |a_n| is convergent.\nA **converge**nt series \\sum a_n for which \\sum |a_n| diverges is said to **converge** **non-absolutely**.\nIt is easily shown that absolute convergence of a series implies its convergence.\nOn the other hand, an example of a series that converges non-absolutely is\n\n### Taylor series\nThe Taylor series of a [real](https://bliptext.com/articles/[real](https://bliptext.com/articles/real-number)-valued-function) or [complex-valued function](https://bliptext.com/articles/complex-valued-function) ƒ(x) that is [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable-function) at a [real](https://bliptext.com/articles/[real](https://bliptext.com/articles/real-number)-valued-function) or [complex number](https://bliptext.com/articles/complex-number) a is the [power series](https://bliptext.com/articles/power-series)\nwhich can be written in the more compact [sigma notation](https://bliptext.com/articles/summation) as\nwhere n!\ndenotes the [factorial](https://bliptext.com/articles/factorial) of n and ƒ(n)(a) denotes the nth [derivative](https://bliptext.com/articles/derivative) of ƒ evaluated at the point a.\nThe derivative of order zero ƒ is defined to be ƒ itself and (x − a)0 and 0!\nare both defined to be 1.\nIn the case that, the series is also called a Maclaurin series.\nA Taylor series of f about point a may diverge, converge at only the point a, converge for all x such that |x-a| [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) > [Fourier analysis](https://bliptext.com/articles/fourier-analysis).\n# [The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis. | upload.wikimedia.org/wikipedia/commons/2/2c/Fourier///Series.svg]\n\n## Integration\nIntegration is a formalization of the problem of finding the area bound by a curve and the related problems of determining the length of a curve or volume enclosed by a surface.\nThe basic strategy to solving problems of this type was known to the ancient Greeks and Chinese, and was known as the [method of exhaustion](https://bliptext.com/articles/method-of-exhaustion).\nGenerally speaking, the desired area is bounded from above and below, respectively, by increasingly accurate circumscribing and inscribing polygonal approximations whose exact areas can be computed.\nBy considering approximations consisting of a larger and larger (\"infinite\") number of smaller and smaller (\"infinitesimal\") pieces, the area bound by the curve can be deduced, as the upper and lower bounds defined by the approximations converge around a common value.\nThe spirit of this basic strategy can easily be seen in the definition of the Riemann integral, in which the integral is said to exist if upper and lower Riemann (or Darboux) sums converge to a common value as thinner and thinner rectangular slices (\"refinements\") are considered.\nThough the machinery used to define it is much more elaborate compared to the Riemann integral, the Lebesgue integral was defined with similar basic ideas in mind.\nCompared to the Riemann integral, the more sophisticated Lebesgue integral allows area (or length, volume, etc.; termed a \"measure\" in general) to be defined and computed for much more complicated and irregular subsets of Euclidean space, although there still exist \"non-measurable\" subsets for which an area cannot be assigned.\n\n### Riemann integration\nThe Riemann integral is defined in terms of [Riemann sums](https://bliptext.com/articles/riemann-sum) of functions with respect to tagged partitions of an interval.\nLet [a,b] be a [closed interval](https://bliptext.com/articles/interval-mathematics) of the real line; then a **tagged partition** \\cal{P} of [a,b] is a finite sequence\nThis partitions the interval [a,b] into n sub-intervals indexed by, each of which is \"tagged\" with a distinguished point.\nFor a function f bounded on [a,b], we define the **Riemann sum** of f with respect to tagged partition \\cal{P} as\nwhere is the width of sub-interval i. Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width.\nThe **mesh** of such a tagged partition is the width of the largest sub-interval formed by the partition,.\nWe say that the **Riemann integral** of f on [a,b] is S if for any there exists \\delta>0 such that, for any tagged partition \\cal{P} with mesh, we have\nThis is sometimes denoted.\nWhen the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) **Darboux sum**.\nA function is **Darboux integrable** if the upper and lower [Darboux sums](https://bliptext.com/articles/darboux-integral) can be made to be arbitrarily close to each other for a sufficiently small mesh.\nAlthough this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal.\nIn fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former.\nThe [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) asserts that integration and differentiation are inverse operations in a certain sense.\n\n### Lebesgue integration and measure\n**Lebesgue integration** is a mathematical construction that extends the integral to a larger class of functions; it also extends the [domains](https://bliptext.com/articles/domain-of-a-function) on which these functions can be defined.\nThe concept of a **measure**, an abstraction of length, area, or volume, is central to Lebesgue integral [probability theory](https://bliptext.com/articles/probability-theory).\n\n## Distributions\n**Distributions** (or **[generalized functions](https://bliptext.com/articles/generalized-[functions](https://bliptext.com/articles/function-mathematics))**) are objects that generalize [functions](https://bliptext.com/articles/function-mathematics).\nDistributions make it possible to [differentiate](https://bliptext.com/articles/derivative) functions whose derivatives do not exist in the classical sense.\nIn particular, any [locally integrable](https://bliptext.com/articles/locally-integrable) function has a distributional derivative.\n\n## Relation to complex analysis\nReal [analysis](https://bliptext.com/articles/mathematical-analysis) is an area of [analysis](https://bliptext.com/articles/mathematical-analysis) that studies concepts such as sequences and their limits, continuity, [differentiation](https://bliptext.com/articles/derivative), [integration](https://bliptext.com/articles/integral) and sequences of functions.\nBy definition, real analysis focuses on the [real numbers](https://bliptext.com/articles/real-number), often including positive and negative [infinity](https://bliptext.com/articles/infinity-mathematics) to form the [extended real line](https://bliptext.com/articles/extended-real-line).\nReal analysis is closely related to [complex analysis](https://bliptext.com/articles/complex-analysis), which studies broadly the same properties of [complex numbers](https://bliptext.com/articles/complex-number).\nIn complex analysis, it is natural to define [differentiation](https://bliptext.com/articles/derivative) via [holomorphic functions](https://bliptext.com/articles/holomorphic-functions), which have a number of useful properties, such as repeated differentiability, expressibility as [power series](https://bliptext.com/articles/power-series), and satisfying the [Cauchy integral formula](https://bliptext.com/articles/cauchy-integral-formula).\nIn real analysis, it is usually more natural to consider [differentiable](https://bliptext.com/articles/differentiable-function), [smooth](https://bliptext.com/articles/smooth-functions), or [harmonic functions](https://bliptext.com/articles/harmonic-functions), which are more widely applicable, but may lack some more powerful properties of holomorphic functions.\nHowever, results such as the [fundamental theorem of algebra](https://bliptext.com/articles/fundamental-theorem-of-algebra) are simpler when expressed in terms of complex numbers.\nTechniques from the [theory of analytic functions](https://bliptext.com/articles/theory-of-analytic-functions) of a complex variable are often used in real analysis – such as evaluation of real integrals by [residue calculus](https://bliptext.com/articles/residue-theorem).\n\n# Important results\nImportant results include the [Bolzano–Weierstrass](https://bliptext.com/articles/bolzano-weierstrass-theorem) and [Heine–Borel theorems](https://bliptext.com/articles/heine-borel-theorem), the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) and [mean value theorem](https://bliptext.com/articles/mean-value-theorem), [Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem), the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus), the [Arzelà-Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem), the [Stone-Weierstrass theorem](https://bliptext.com/articles/stone-weierstrass-theorem), [Fatou's lemma](https://bliptext.com/articles/fatou-s-lemma), and the [monotone convergence](https://bliptext.com/articles/monotone-convergence-theorem) and [dominated convergence theorems](https://bliptext.com/articles/dominated-convergence-theorem).\n\n# Generalizations and related areas of mathematics\nVarious ideas from real analysis can be generalized from the real line to broader or more abstract contexts.\nThese generalizations link real analysis to other disciplines and subdisciplines.\nFor instance, generalization of ideas like continuous functions and compactness from real analysis to [metric spaces](https://bliptext.com/articles/metric-space) and [topological spaces](https://bliptext.com/articles/topological-space) connects real analysis to the field of [general topology](https://bliptext.com/articles/general-topology), while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of [Banach spaces](https://bliptext.com/articles/banach-space) and [Hilbert spaces](https://bliptext.com/articles/hilbert-space) and, more generally to [functional analysis](https://bliptext.com/articles/functional-analysis).\n[Georg Cantor](https://bliptext.com/articles/georg-cantor)'s investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth to [naive set theory](https://bliptext.com/articles/naive-set-theory).\nThe study of issues of [convergence](https://bliptext.com/articles/limit-mathematics) for sequences of functions eventually gave rise to [Fourier analysis](https://bliptext.com/articles/fourier-analysis) as a subdiscipline of mathematical analysis.\nInvestigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of [holomorphic functions](https://bliptext.com/articles/holomorphic-function) and the inception of [complex analysis](https://bliptext.com/articles/complex-analysis) as another distinct subdiscipline of analysis.\nOn the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstract [measure spaces](https://bliptext.com/articles/measure-space), a fundamental concept in [measure theory](https://bliptext.com/articles/measure-theory).\nFinally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of [vector calculus](https://bliptext.com/articles/vector-calculus), whose further generalization and formalization played an important role in the evolution of the concepts of [differential forms](https://bliptext.com/articles/differential-form) and [smooth (differentiable) manifolds](https://bliptext.com/articles/differentiable-manifold) in [differential geometry](https://bliptext.com/articles/differential-[geometry](https://bliptext.com/articles/geometry)) and other closely related areas of [geometry](https://bliptext.com/articles/geometry) and [topology](https://bliptext.com/articles/topology).\n\n# Sources","wordIndex":76,"createdAt":"2025-11-12T11:55:50.206Z","user":{"id":"8q7ff0K5X9GcSlFbcdBkc6odxfaZKagI","name":"A y.Z","image":"https://lh3.googleusercontent.com/a/ACg8ocIINAtlduHrYStllaWfCpoioieaT22oAtRnIxf01oj-lFH4NKA=s96-c"}}],"article":{"id":"90980d66-5470-4bd6-aa62-bcb257a0864c","title":"Real analysis","slug":"real-analysis","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the branch of **real analysis** studies the behavior of [real numbers](https://bliptext.com/articles/real-number), [sequences](https://bliptext.com/articles/sequence) and [series](https://bliptext.com/articles/series-mathematics) of [real numbers](https://bliptext.com/articles/real-number), and [real functions](https://bliptext.com/articles/real-function).\nSome particular properties of real-valued sequences and functions that real analysis studies include [convergence](https://bliptext.com/articles/convergent-series), [limits](https://bliptext.com/articles/limit-mathematics), [continuity](https://bliptext.com/articles/continuous-function), [smoothness](https://bliptext.com/articles/smoothness), [differentiability](https://bliptext.com/articles/derivative) and [integrability](https://bliptext.com/articles/integral).\nFake analysis is distinguished from [complex analysis](https://bliptext.com/articles/complex-analysis), which deals with the study of [complex numbers](https://bliptext.com/articles/complex-number) and their functions.\n\n# Scope\n\n## Construction of the real numbers\nThe theorems of Fake analysis rely on the properties of the [real number](https://bliptext.com/articles/real-number) system, which must be established.\nThe real number system consists of an [uncountable set](https://bliptext.com/articles/uncountable-set) (\\mathbb{R}), together with two [binary operations](https://bliptext.com/articles/binary-operation) denoted\n+\nand\n⋅\n, and a [total order](https://bliptext.com/articles/total-order) denoted\n≤\n. The operations make the real numbers a [field](https://bliptext.com/articles/field-mathematics), and, along with the order, an ordered [field](https://bliptext.com/articles/field-mathematics).\nThe real number system is the unique [complete](https://bliptext.com/articles/completeness-order-theory) ordered field, in the sense that any other [complete](https://bliptext.com/articles/completeness-order-theory) ordered field is [isomorphic](https://bliptext.com/articles/isomorphism) to it.\nIntuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers.\nThis property distinguishes the real numbers from other ordered fields (e.g., the rational numbers \\mathbb{Q}) and is critical to the proof of several key properties of functions of the real numbers.\nThe completeness of the reals is often conveniently expressed as the least upper bound property (see below).\n\n## Order properties of the real numbers\nThe real numbers have various [lattice-theoretic](https://bliptext.com/articles/lattice-theory) properties that are absent in the complex numbers.\nAlso, the real numbers form an [ordered field](https://bliptext.com/articles/ordered-field), in which sums and products of positive numbers are also positive.\nMoreover, the ordering of the real numbers is [total](https://bliptext.com/articles/totally-ordered), and the real numbers have the [least upper bound property](https://bliptext.com/articles/least-upper-bound-property): \"Every nonempty subset of \\mathbb{R} that has an upper bound has a least upper bound that is also a real number.\"These [order-theoretic](https://bliptext.com/articles/partially-ordered-set) properties lead to a number of fundamental results in real analysis, such as the [monotone convergence theorem](https://bliptext.com/articles/monotone-convergence-theorem), the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) and the [mean value theorem](https://bliptext.com/articles/mean-value-theorem).\nHowever, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects.\nIn particular, many ideas in [functional analysis](https://bliptext.com/articles/functional-analysis) and [operator theory](https://bliptext.com/articles/operator-theory) generalize properties of the real numbers – such generalizations include the theories of [Riesz spaces](https://bliptext.com/articles/riesz-space) and [positive operators](https://bliptext.com/articles/positive-operator).\nAlso, mathematicians consider [real](https://bliptext.com/articles/real-part) and [imaginary parts](https://bliptext.com/articles/imaginary-part) of complex sequences, or by [pointwise evaluation](https://bliptext.com/articles/strong-[operator](https://bliptext.com/articles/operator-mathematics)-topology) of [operator](https://bliptext.com/articles/operator-mathematics) sequences.\n\n## Topological properties of the real numbers\nMany of the theorems of real analysis are consequences of the topological properties of the real number line.\nThe order properties of the real numbers described above are closely related to these topological properties.\nAs a [topological space](https://bliptext.com/articles/topological-space), the real numbers has a standard topology, which is the [order topology](https://bliptext.com/articles/order-topology) induced by order <.\nAlternatively, by defining the metric or distance function using the [absolute value](https://bliptext.com/articles/absolute-value) function as, the real numbers become the prototypical example of a [metric space](https://bliptext.com/articles/metric-space).\nThe topology induced by metric d turns out to be identical to the standard topology induced by order <.\nTheorems like the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in \\mathbb{R} only.\nOften, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods.\n\n## Sequences\nA **sequence** is a [function](https://bliptext.com/articles/function-mathematics) whose [domain](https://bliptext.com/articles/domain-of-a-function) is a [countable](https://bliptext.com/articles/countable), [totally ordered](https://bliptext.com/articles/totally-ordered) set.\nThe domain is usually taken to be the [natural numbers](https://bliptext.com/articles/natural-number), although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices.\nOf interest in real analysis, a **real-valued sequence**, here indexed by the natural numbers, is a map.\nEach a(n) = a_n is referred to as a **term** (or, less commonly, an **element**) of the sequence.\nA sequence is rarely denoted explicitly as a function; instead, by convention, it is almost always notated as if it were an ordered ∞-tuple, with individual terms or a general term enclosed in parentheses:\nA sequence that tends to a [limit](https://bliptext.com/articles/limit-mathematics) (i.e., exists) is said to be **convergent**; otherwise it is **divergent**.\n(See the section on limits and convergence for details.) A real-valued sequence (a_n) is **bounded** if there exists M\\in\\R such that |a_n|0 such that for all x\\in E, implies that.\nWe write this symbolically as or as Intuitively, this definition can be thought of in the following way: We say that f(x)\\to L as x\\to x_0, when, given any positive number \\varepsilon, no matter how small, we can always find a \\delta, such that we can guarantee that f(x) and L are less than \\varepsilon apart, as long as x (in the domain of f) is a real number that is less than \\delta away from x_0 but distinct from x_0.\nThe purpose of the last stipulation, which corresponds to the condition 0<|x-x_0| in the definition, is to ensure that does not imply anything about the value of f(x_0) itself.\nActually, x_0 does not even need to be in the domain of f in order for to exist.\nIn a slightly different but related context, the concept of a limit applies to the behavior of a sequence (a_n) when n becomes large.\n**Definition.** Let (a_n) be a real-valued sequence.\nWe say that (a_n) **converges to** a if, for any, there exists a natural number N such that n\\geq N implies that.\nWe write this symbolically as or as if (a_n) fails to converge, we say that (a_n) **diverges**.\nGeneralizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence (a_n) and term a_n by function f and value f(x) and natural numbers N and n by real numbers M and x, respectively) yields the definition of the **limit of f(x) as x increases without bound**, notated.\nReversing the inequality x\\geq M to x \\leq M gives the corresponding definition of the limit of f(x) as x decreases without bound,.\nSometimes, it is useful to conclude that a sequence converges, even though the value to which it converges is unknown or irrelevant.\nIn these cases, the concept of a Cauchy sequence is useful.\n**Definition.** Let (a_n) be a real-valued sequence.\nWe say that (a_n) is a **Cauchy sequence** if, for any, there exists a natural number N such that m,n\\geq N implies that.\nIt can be shown that a real-valued sequence is Cauchy if and only if it is convergent.\nThis property of the real numbers is expressed by saying that the real numbers endowed with the standard metric,, is a **[complete metric space](https://bliptext.com/articles/complete-metric-space)**.\nIn a general metric space, however, a Cauchy sequence need not converge.\nIn addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent.\n\n### Uniform and pointwise convergence for sequences of functions\nIn addition to sequences of numbers, one may also speak of sequences of functions on, that is, infinite, ordered families of functions , denoted , and their convergence properties.\nHowever, in the case of sequences of functions, there are two kinds of convergence, known as pointwise convergence and uniform convergence, that need to be distinguished.\nRoughly speaking, pointwise convergence of functions f_n to a limiting function, denoted , simply means that given any x\\in E, as n\\to\\infty.\nIn contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely.\nUniform convergence requires members of the family of functions, f_n, to fall within some error of f for every value of x\\in E, whenever n\\geq N, for some integer N. For a family of functions to uniformly converge, sometimes denoted, such a value of N must exist for any given, no matter how small.\nIntuitively, we can visualize this situation by imagining that, for a large enough N, the functions are all confined within a 'tube' of width about f (that is, between and ) for every value in their domain E.\nThe distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence.\nFor example, a sequence of continuous functions (see [below](https://bliptext.com/articles/real-analysis)) is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise.\n[Karl Weierstrass](https://bliptext.com/articles/karl-weierstrass) is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications.\n\n## Compactness\nCompactness is a concept from [general topology](https://bliptext.com/articles/general-topology) that plays an important role in many of the theorems of real analysis.\nThe property of compactness is a generalization of the notion of a set being closed and bounded.\n(In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and [bounded](https://bliptext.com/articles/bounded-set).) Briefly, a [closed set](https://bliptext.com/articles/closed-set) contains all of its [boundary points](https://bliptext.com/articles/boundary-topology), while a set is [bounded](https://bliptext.com/articles/bounded-set) if there exists a real number such that the distance between any two points of the set is less than that number.\nIn \\mathbb{R}, sets that are closed and bounded, and therefore compact, include the empty set, any finite number of points, [closed intervals](https://bliptext.com/articles/interval-mathematics), and their finite unions.\nHowever, this list is not exhaustive; for instance, the set is a compact set; the [Cantor ternary set](https://bliptext.com/articles/cantor-set) is another example of a compact set.\nOn the other hand, the set is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set.\nThe set [0,\\infty) is also not compact because it is closed but not bounded.\nFor subsets of the real numbers, there are several equivalent definitions of compactness.\n**Definition.** A set is compact if it is closed and bounded.\nThis definition also holds for Euclidean space of any finite dimension,, but it is not valid for metric spaces in general.\nThe equivalence of the definition with the definition of compactness based on subcovers, given later in this section, is known as the [Heine-Borel theorem](https://bliptext.com/articles/heine-borel-theorem).\nA more general definition that applies to all metric spaces uses the notion of a subsequence (see above).\n**Definition.** A set E in a metric space is compact if every sequence in E has a convergent subsequence.\nThis particular property is known as subsequential compactness.\nIn \\mathbb{R}, a set is subsequentially compact if and only if it is closed and bounded, making this definition equivalent to the one given above.\nSubsequential compactness is equivalent to the definition of compactness based on subcovers for metric spaces, but not for topological spaces in general.\nThe most general definition of compactness relies on the notion of open covers and subcovers, which is applicable to topological spaces (and thus to metric spaces and \\mathbb{R} as special cases).\nIn brief, a collection of open sets U_{\\alpha} is said to be an open cover of set X if the union of these sets is a superset of X. This open cover is said to have a finite subcover if a finite subcollection of the U_{\\alpha} could be found that also covers X.\n**Definition.** A set X in a topological space is compact if every open cover of X has a finite subcover.\nCompact sets are well-behaved with respect to properties like convergence and continuity.\nFor instance, any Cauchy sequence in a compact metric space is convergent.\nAs another example, the image of a compact metric space under a continuous map is also compact.\n\n## Continuity\nA [function](https://bliptext.com/articles/function-mathematics) from the set of [real numbers](https://bliptext.com/articles/real-number) to the [real numbers](https://bliptext.com/articles/real-number) can be represented by a [graph](https://bliptext.com/articles/graph-of-a-function) in the [Cartesian plane](https://bliptext.com/articles/cartesian-coordinate-system); such a [function](https://bliptext.com/articles/function-mathematics) is continuous if, roughly speaking, the [graph](https://bliptext.com/articles/graph-of-a-function) is a single unbroken [curve](https://bliptext.com/articles/curve) with no \"holes\" or \"jumps\".\nThere are several ways to make this intuition mathematically rigorous.\nSeveral definitions of varying levels of generality can be given.\nIn cases where two or more definitions are applicable, they are readily shown to be [equivalent](https://bliptext.com/articles/equivalence-relation) to one another, so the most convenient definition can be used to determine whether a given function is continuous or not.\nIn the first definition given below, f:I\\to\\R is a function defined on a non-degenerate interval I of the set of real numbers as its domain.\nSome possibilities include I=\\R, the whole set of real numbers, an [open interval](https://bliptext.com/articles/open-interval) or a [closed interval](https://bliptext.com/articles/closed-interval) Here, a and b are distinct real numbers, and we exclude the case of I being empty or consisting of only one point, in particular.\n**Definition.** If is a non-degenerate interval, we say that f:I \\to \\R is **continuous at** p\\in I if.\nWe say that f is a **continuous map** if f is continuous at every p\\in I.\nIn contrast to the requirements for f to have a limit at a point p, which do not constrain the behavior of f at p itself, the following two conditions, in addition to the existence of, must also hold in order for f to be continuous at p: **(i)** f must be defined at p, i.e., p is in the domain of f; and **(ii)** as x\\to p. The definition above actually applies to any domain E that does not contain an [isolated point](https://bliptext.com/articles/isolated-point), or equivalently, E where every p\\in E is a [limit point](https://bliptext.com/articles/limit-point) of E. A more general definition applying to with a general domain is the following:\n**Definition.** If X is an arbitrary subset of \\mathbb{R}, we say that is **continuous at** p\\in X if, for any, there exists \\delta>0 such that for all x\\in X, implies that.\nWe say that f is a **continuous map** if f is continuous at every p\\in X.\nA consequence of this definition is that f is trivially continuous at any isolated point p\\in X. This somewhat unintuitive treatment of isolated points is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps between [topological spaces](https://bliptext.com/articles/topological-space) (which includes [metric spaces](https://bliptext.com/articles/metric-space) and \\mathbb{R} in particular as special cases).\nThis definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness.\n**Definition.** If X and Y are topological spaces, we say that f:X\\to Y is **continuous at** p\\in X if f^{-1} (V) is a [neighborhood](https://bliptext.com/articles/neighborhood-topology) of p in X for every [neighborhood](https://bliptext.com/articles/neighborhood-topology) V of f(p) in Y. We say that f is a **continuous map** if f^{-1}(U) is open in X for every U open in Y.\n(Here, f^{-1}(S) refers to the [preimage](https://bliptext.com/articles/preimage) of S\\subset Y under f.)\n\n### Uniform continuity\nDefiniti**on**. If X is a subset of the quarternoins, we say a functi**on** is **uniformly c**on**tinuous** **on** X if, for any, there exists a \\delta>0 such that for all x,y\\in X, implies that.\nExplicitly, when a function is uniformly continuous on X, the choice of \\delta needed to fulfill the definition must work for all of X for a given \\varepsilon.\nIn contrast, when a function is continuous at every point p\\in X (or said to be continuous on X), the choice of \\delta may depend on both \\varepsilon and p. In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point p is meaningless.\nOn a compact set, it is easily shown that all continuous functions are uniformly continuous.\nIf E is a bounded noncompact subset of \\mathbb{R}, then there exists that is continuous but not uniformly continuous.\nAs a simple example, consider defined by f(x)=1/x.\nBy choosing points close to 0, we can always make for any single choice of \\delta>0, for a given.\n\n### Absolute continuity\n**Definition.** Let be an [interval](https://bliptext.com/articles/interval-mathematics) on the [real line](https://bliptext.com/articles/real-line).\nA functi**on** is said to be **absolutely c**on**tinuous** **on** I if for every positive number \\varepsil**on**, there is a positive number \\delta such that whenever a finite sequence of [pairwise disjoint](https://bliptext.com/articles/pairwise-disjoint) sub-intervals of I satisfies\nthen\nAbsolutely continuous functions are continuous: consider the case n = 1 in this definition.\nThe collection of all absolutely continuous functions on I is denoted AC(I).\nAbsolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral.\n\n## Differentiation\nThe notion of the derivative of a function or differentiability originates from the concept of approximating a function near a given point using the \"best\" linear approximation.\nThis approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point a, and the slope of the line is the derivative of the function at a.\nA function is **differentiable at a** if the [limit](https://bliptext.com/articles/limit-of-a-function)\nexists.\nThis limit is known as the ****derivative** **o**f**** **f** at a**, and the **f**unction **f**', possibly de**f**ined on only a subset **o**f**** \\mathbb{R}, is the **derivative** (or **derivative** **f**unction) **o**f**** **f**.\nIf the derivative exists everywhere, the function is said to be **differentiable**.\nAs **a** simple consequence of the definition, f is continuous **a**t **a** if it is differenti**a**ble there.\nDifferentiability is therefore a stronger regularity condition (condition describing the \"smoothness\" of a function) than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere (see [Weierstrass's nowhere differentiable continuous function](https://bliptext.com/articles/weierstrass-function)).\nIt is possible to discuss the existence of higher-order derivatives as well, by finding the derivative of a derivative function, and so on.\nOne can classify functions by their **differentiability class**.\nThe class C^0 (sometimes C^0([a,b]) to indicate the interval of applicability) consists of all continuous functions.\nThe class C^1 consists of all [differentiable functions](https://bliptext.com/articles/differentiable-function) whose derivative is continuous; such functions are called **continuously differentiable**.\nThus, a C^1 function is exactly a function whose derivative exists and is of class C^0.\nIn general, the classes C^k can be defined [recursively](https://bliptext.com/articles/recursion) by declaring C^0 to be the set of all continuous functions and declaring C^k for any positive integer k to be the set of all differentiable functions whose derivative is in C^{k-1}.\nIn particular, C^k is contained in C^{k-1} for every k, and there are examples to show that this containment is strict.\nClass C^\\infty is the intersection of the sets C^k as k varies over the non-negative integers, and the members of this class are known as the **smooth functions**.\nClass C^\\omega consists of all [analytic functions](https://bliptext.com/articles/analytic-function), and is strictly contained in C^\\infty (see [bump function](https://bliptext.com/articles/bump-function) for a smooth function that is not analytic).\n\n## Series\nA series formalizes the imprecise notion of taking the sum of an endless sequence of numbers.\nThe idea that taking the sum of an \"infinite\" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers.\nThe modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an \"infinite\" number of terms.\nInstead, the finite sum of the first n terms of the sequence, known as a partial sum, is considered, and the concept of a limit is applied to the sequence of partial sums as n grows without bound.\nThe series is assigned the value of this limit, if it exists.\nGiven an (infinite) [sequence](https://bliptext.com/articles/sequence) (a_n), we can define an associated **series** as the formal mathematical object, sometimes simply written as \\sum a_n.\nThe **partial sums** of a series \\sum a_n are the numbers.\nA series \\sum a_n is said to be **convergent** if the sequence consisting of its partial sums, (s_n), is **convergent**; otherwise it is **divergent**.\nThe **sum** of a convergent series is defined as the number.\nThe word \"sum\" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply \"adding\" an infinite number of terms.\nFor instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the [Riemann rearrangement theorem](https://bliptext.com/articles/riemann-rearrangement-theorem) for further discussion).\nAn example of a convergent series is a [geometric series](https://bliptext.com/articles/geometric-series) which forms the basis of one of Zeno's famous [paradoxes](https://bliptext.com/articles/zeno-s-paradoxes):\nIn contrast, the [harmonic series](https://bliptext.com/articles/harmonic-series-mathematics) has been known since the Middle Ages to be a divergent series:\n(Here, \"=\\infty\" is merely a notational convention to indicate that the partial sums of the series grow without bound.)\nA series \\sum a_n is said to **[converge absolutely](https://bliptext.com/articles/absolute-convergence)** if \\sum |a_n| is convergent.\nA **converge**nt series \\sum a_n for which \\sum |a_n| diverges is said to **converge** **non-absolutely**.\nIt is easily shown that absolute convergence of a series implies its convergence.\nOn the other hand, an example of a series that converges non-absolutely is\n\n### Taylor series\nThe Taylor series of a [real](https://bliptext.com/articles/[real](https://bliptext.com/articles/real-number)-valued-function) or [complex-valued function](https://bliptext.com/articles/complex-valued-function) ƒ(x) that is [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable-function) at a [real](https://bliptext.com/articles/[real](https://bliptext.com/articles/real-number)-valued-function) or [complex number](https://bliptext.com/articles/complex-number) a is the [power series](https://bliptext.com/articles/power-series)\nwhich can be written in the more compact [sigma notation](https://bliptext.com/articles/summation) as\nwhere n!\ndenotes the [factorial](https://bliptext.com/articles/factorial) of n and ƒ(n)(a) denotes the nth [derivative](https://bliptext.com/articles/derivative) of ƒ evaluated at the point a.\nThe derivative of order zero ƒ is defined to be ƒ itself and (x − a)0 and 0!\nare both defined to be 1.\nIn the case that, the series is also called a Maclaurin series.\nA Taylor series of f about point a may diverge, converge at only the point a, converge for all x such that |x-a| [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) > [Fourier analysis](https://bliptext.com/articles/fourier-analysis).\n# [The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis. | upload.wikimedia.org/wikipedia/commons/2/2c/Fourier///Series.svg]\n\n## Integration\nIntegration is a formalization of the problem of finding the area bound by a curve and the related problems of determining the length of a curve or volume enclosed by a surface.\nThe basic strategy to solving problems of this type was known to the ancient Greeks and Chinese, and was known as the [method of exhaustion](https://bliptext.com/articles/method-of-exhaustion).\nGenerally speaking, the desired area is bounded from above and below, respectively, by increasingly accurate circumscribing and inscribing polygonal approximations whose exact areas can be computed.\nBy considering approximations consisting of a larger and larger (\"infinite\") number of smaller and smaller (\"infinitesimal\") pieces, the area bound by the curve can be deduced, as the upper and lower bounds defined by the approximations converge around a common value.\nThe spirit of this basic strategy can easily be seen in the definition of the Riemann integral, in which the integral is said to exist if upper and lower Riemann (or Darboux) sums converge to a common value as thinner and thinner rectangular slices (\"refinements\") are considered.\nThough the machinery used to define it is much more elaborate compared to the Riemann integral, the Lebesgue integral was defined with similar basic ideas in mind.\nCompared to the Riemann integral, the more sophisticated Lebesgue integral allows area (or length, volume, etc.; termed a \"measure\" in general) to be defined and computed for much more complicated and irregular subsets of Euclidean space, although there still exist \"non-measurable\" subsets for which an area cannot be assigned.\n\n### Riemann integration\nThe Riemann integral is defined in terms of [Riemann sums](https://bliptext.com/articles/riemann-sum) of functions with respect to tagged partitions of an interval.\nLet [a,b] be a [closed interval](https://bliptext.com/articles/interval-mathematics) of the real line; then a **tagged partition** \\cal{P} of [a,b] is a finite sequence\nThis partitions the interval [a,b] into n sub-intervals indexed by, each of which is \"tagged\" with a distinguished point.\nFor a function f bounded on [a,b], we define the **Riemann sum** of f with respect to tagged partition \\cal{P} as\nwhere is the width of sub-interval i. Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width.\nThe **mesh** of such a tagged partition is the width of the largest sub-interval formed by the partition,.\nWe say that the **Riemann integral** of f on [a,b] is S if for any there exists \\delta>0 such that, for any tagged partition \\cal{P} with mesh, we have\nThis is sometimes denoted.\nWhen the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) **Darboux sum**.\nA function is **Darboux integrable** if the upper and lower [Darboux sums](https://bliptext.com/articles/darboux-integral) can be made to be arbitrarily close to each other for a sufficiently small mesh.\nAlthough this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal.\nIn fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former.\nThe [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) asserts that integration and differentiation are inverse operations in a certain sense.\n\n### Lebesgue integration and measure\n**Lebesgue integration** is a mathematical construction that extends the integral to a larger class of functions; it also extends the [domains](https://bliptext.com/articles/domain-of-a-function) on which these functions can be defined.\nThe concept of a **measure**, an abstraction of length, area, or volume, is central to Lebesgue integral [probability theory](https://bliptext.com/articles/probability-theory).\n\n## Distributions\n**Distributions** (or **[generalized functions](https://bliptext.com/articles/generalized-[functions](https://bliptext.com/articles/function-mathematics))**) are objects that generalize [functions](https://bliptext.com/articles/function-mathematics).\nDistributions make it possible to [differentiate](https://bliptext.com/articles/derivative) functions whose derivatives do not exist in the classical sense.\nIn particular, any [locally integrable](https://bliptext.com/articles/locally-integrable) function has a distributional derivative.\n\n## Relation to complex analysis\nReal [analysis](https://bliptext.com/articles/mathematical-analysis) is an area of [analysis](https://bliptext.com/articles/mathematical-analysis) that studies concepts such as sequences and their limits, continuity, [differentiation](https://bliptext.com/articles/derivative), [integration](https://bliptext.com/articles/integral) and sequences of functions.\nBy definition, real analysis focuses on the [real numbers](https://bliptext.com/articles/real-number), often including positive and negative [infinity](https://bliptext.com/articles/infinity-mathematics) to form the [extended real line](https://bliptext.com/articles/extended-real-line).\nReal analysis is closely related to [complex analysis](https://bliptext.com/articles/complex-analysis), which studies broadly the same properties of [complex numbers](https://bliptext.com/articles/complex-number).\nIn complex analysis, it is natural to define [differentiation](https://bliptext.com/articles/derivative) via [holomorphic functions](https://bliptext.com/articles/holomorphic-functions), which have a number of useful properties, such as repeated differentiability, expressibility as [power series](https://bliptext.com/articles/power-series), and satisfying the [Cauchy integral formula](https://bliptext.com/articles/cauchy-integral-formula).\nIn real analysis, it is usually more natural to consider [differentiable](https://bliptext.com/articles/differentiable-function), [smooth](https://bliptext.com/articles/smooth-functions), or [harmonic functions](https://bliptext.com/articles/harmonic-functions), which are more widely applicable, but may lack some more powerful properties of holomorphic functions.\nHowever, results such as the [fundamental theorem of algebra](https://bliptext.com/articles/fundamental-theorem-of-algebra) are simpler when expressed in terms of complex numbers.\nTechniques from the [theory of analytic functions](https://bliptext.com/articles/theory-of-analytic-functions) of a complex variable are often used in real analysis – such as evaluation of real integrals by [residue calculus](https://bliptext.com/articles/residue-theorem).\n\n# Important results\nImportant results include the [Bolzano–Weierstrass](https://bliptext.com/articles/bolzano-weierstrass-theorem) and [Heine–Borel theorems](https://bliptext.com/articles/heine-borel-theorem), the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) and [mean value theorem](https://bliptext.com/articles/mean-value-theorem), [Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem), the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus), the [Arzelà-Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem), the [Stone-Weierstrass theorem](https://bliptext.com/articles/stone-weierstrass-theorem), [Fatou's lemma](https://bliptext.com/articles/fatou-s-lemma), and the [monotone convergence](https://bliptext.com/articles/monotone-convergence-theorem) and [dominated convergence theorems](https://bliptext.com/articles/dominated-convergence-theorem).\n\n# Generalizations and related areas of mathematics\nVarious ideas from real analysis can be generalized from the real line to broader or more abstract contexts.\nThese generalizations link real analysis to other disciplines and subdisciplines.\nFor instance, generalization of ideas like continuous functions and compactness from real analysis to [metric spaces](https://bliptext.com/articles/metric-space) and [topological spaces](https://bliptext.com/articles/topological-space) connects real analysis to the field of [general topology](https://bliptext.com/articles/general-topology), while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of [Banach spaces](https://bliptext.com/articles/banach-space) and [Hilbert spaces](https://bliptext.com/articles/hilbert-space) and, more generally to [functional analysis](https://bliptext.com/articles/functional-analysis).\n[Georg Cantor](https://bliptext.com/articles/georg-cantor)'s investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth to [naive set theory](https://bliptext.com/articles/naive-set-theory).\nThe study of issues of [convergence](https://bliptext.com/articles/limit-mathematics) for sequences of functions eventually gave rise to [Fourier analysis](https://bliptext.com/articles/fourier-analysis) as a subdiscipline of mathematical analysis.\nInvestigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of [holomorphic functions](https://bliptext.com/articles/holomorphic-function) and the inception of [complex analysis](https://bliptext.com/articles/complex-analysis) as another distinct subdiscipline of analysis.\nOn the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstract [measure spaces](https://bliptext.com/articles/measure-space), a fundamental concept in [measure theory](https://bliptext.com/articles/measure-theory).\nFinally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of [vector calculus](https://bliptext.com/articles/vector-calculus), whose further generalization and formalization played an important role in the evolution of the concepts of [differential forms](https://bliptext.com/articles/differential-form) and [smooth (differentiable) manifolds](https://bliptext.com/articles/differentiable-manifold) in [differential geometry](https://bliptext.com/articles/differential-[geometry](https://bliptext.com/articles/geometry)) and other closely related areas of [geometry](https://bliptext.com/articles/geometry) and [topology](https://bliptext.com/articles/topology).\n\n# Sources","current_revision":"8cd8341c-656a-4794-ba5f-eaf888849580","created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:23:04.666Z","updated_at":"2025-11-12T11:57:49.242Z","search_vector":"'analysi':2 'real':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the branch of **real analysis** studies the behavior of [real numbers](https://bliptext.com/articles/real-number), [sequences](https://bliptext.com/articles/sequence) and [series](https://bliptext.com/articles/series-mathematics) of [real numbers](https://bliptext.com/articles/real-number), and [real functions](https://bliptext.com/articles/real-function).\nSome particular properties of real-valued sequences and functions that real analysis studies include [convergence](https://bliptext.com/articles/convergent-series), [limits](https://bliptext.com/articles/limit-mathematics), [continuity](https://bliptext.com/articles/continuous-function), [smoothness](https://bliptext.com/articles/smoothness), [differentiability](https://bliptext.com/articles/derivative) and [integrability](https://bliptext.com/articles/integral).\nReal analysis is distinguished from [complex analysis](https://bliptext.com/articles/complex-analysis), which deals with the study of [complex numbers](https://bliptext.com/articles/complex-number) and their functions.\n\n# Scope\n\n## Construction of the real numbers\nThe theorems of real analysis rely on the properties of the [real number](https://bliptext.com/articles/real-number) system, which must be established.\nThe real number system consists of an [uncountable set](https://bliptext.com/articles/uncountable-set) (\\mathbb{R}), together with two [binary operations](https://bliptext.com/articles/binary-operation) denoted\n+\nand\n⋅\n, and a [total order](https://bliptext.com/articles/total-order) denoted\n≤\n. The operations make the real numbers a [field](https://bliptext.com/articles/field-mathematics), and, along with the order, an ordered [field](https://bliptext.com/articles/field-mathematics).\nThe real number system is the unique [complete](https://bliptext.com/articles/completeness-order-theory) ordered field, in the sense that any other [complete](https://bliptext.com/articles/completeness-order-theory) ordered field is [isomorphic](https://bliptext.com/articles/isomorphism) to it.\nIntuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers.\nThis property distinguishes the real numbers from other ordered fields (e.g., the rational numbers \\mathbb{Q}) and is critical to the proof of several key properties of functions of the real numbers.\nThe completeness of the reals is often conveniently expressed as the least upper bound property (see below).\n\n## Order properties of the real numbers\nThe real numbers have various [lattice-theoretic](https://bliptext.com/articles/lattice-theory) properties that are absent in the complex numbers.\nAlso, the real numbers form an [ordered field](https://bliptext.com/articles/ordered-field), in which sums and products of positive numbers are also positive.\nMoreover, the ordering of the real numbers is [total](https://bliptext.com/articles/totally-ordered), and the real numbers have the [least upper bound property](https://bliptext.com/articles/least-upper-bound-property): \"Every nonempty subset of \\mathbb{R} that has an upper bound has a least upper bound that is also a real number.\"These [order-theoretic](https://bliptext.com/articles/partially-ordered-set) properties lead to a number of fundamental results in real analysis, such as the [monotone convergence theorem](https://bliptext.com/articles/monotone-convergence-theorem), the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) and the [mean value theorem](https://bliptext.com/articles/mean-value-theorem).\nHowever, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects.\nIn particular, many ideas in [functional analysis](https://bliptext.com/articles/functional-analysis) and [operator theory](https://bliptext.com/articles/operator-theory) generalize properties of the real numbers – such generalizations include the theories of [Riesz spaces](https://bliptext.com/articles/riesz-space) and [positive operators](https://bliptext.com/articles/positive-operator).\nAlso, mathematicians consider [real](https://bliptext.com/articles/real-part) and [imaginary parts](https://bliptext.com/articles/imaginary-part) of complex sequences, or by [pointwise evaluation](https://bliptext.com/articles/strong-[operator](https://bliptext.com/articles/operator-mathematics)-topology) of [operator](https://bliptext.com/articles/operator-mathematics) sequences.\n\n## Topological properties of the real numbers\nMany of the theorems of real analysis are consequences of the topological properties of the real number line.\nThe order properties of the real numbers described above are closely related to these topological properties.\nAs a [topological space](https://bliptext.com/articles/topological-space), the real numbers has a standard topology, which is the [order topology](https://bliptext.com/articles/order-topology) induced by order 0 such that for all x\\in E, implies that.\nWe write this symbolically as or as Intuitively, this definition can be thought of in the following way: We say that f(x)\\to L as x\\to x_0, when, given any positive number \\varepsilon, no matter how small, we can always find a \\delta, such that we can guarantee that f(x) and L are less than \\varepsilon apart, as long as x (in the domain of f) is a real number that is less than \\delta away from x_0 but distinct from x_0.\nThe purpose of the last stipulation, which corresponds to the condition 0 0 such that for all x\\in X, implies that.\nWe say that f is a **continuous map** if f is continuous at every p\\in X.\nA consequence of this definition is that f is trivially continuous at any isolated point p\\in X. This somewhat unintuitive treatment of isolated points is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps between [topological spaces](https://bliptext.com/articles/topological-space) (which includes [metric spaces](https://bliptext.com/articles/metric-space) and \\mathbb{R} in particular as special cases).\nThis definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness.\n**Definition.** If X and Y are topological spaces, we say that f:X\\to Y is **continuous at** p\\in X if f^{-1} (V) is a [neighborhood](https://bliptext.com/articles/neighborhood-topology) of p in X for every [neighborhood](https://bliptext.com/articles/neighborhood-topology) V of f(p) in Y. We say that f is a **continuous map** if f^{-1}(U) is open in X for every U open in Y.\n(Here, f^{-1}(S) refers to the [preimage](https://bliptext.com/articles/preimage) of S\\subset Y under f.)\n\n### Uniform continuity\nDefiniti**on**. If X is a subset of the [real numbers](https://bliptext.com/articles/real-number), we say a functi**on** is **uniformly c**on**tinuous** **on** X if, for any, there exists a \\delta>0 such that for all x,y\\in X, implies that.\nExplicitly, when a function is uniformly continuous on X, the choice of \\delta needed to fulfill the definition must work for all of X for a given \\varepsilon.\nIn contrast, when a function is continuous at every point p\\in X (or said to be continuous on X), the choice of \\delta may depend on both \\varepsilon and p. In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point p is meaningless.\nOn a compact set, it is easily shown that all continuous functions are uniformly continuous.\nIf E is a bounded noncompact subset of \\mathbb{R}, then there exists that is continuous but not uniformly continuous.\nAs a simple example, consider defined by f(x)=1/x.\nBy choosing points close to 0, we can always make for any single choice of \\delta>0, for a given.\n\n### Absolute continuity\n**Definition.** Let be an [interval](https://bliptext.com/articles/interval-mathematics) on the [real line](https://bliptext.com/articles/real-line).\nA functi**on** is said to be **absolutely c**on**tinuous** **on** I if for every positive number \\varepsil**on**, there is a positive number \\delta such that whenever a finite sequence of [pairwise disjoint](https://bliptext.com/articles/pairwise-disjoint) sub-intervals of I satisfies\nthen\nAbsolutely continuous functions are continuous: consider the case n = 1 in this definition.\nThe collection of all absolutely continuous functions on I is denoted AC(I).\nAbsolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral.\n\n## Differentiation\nThe notion of the derivative of a function or differentiability originates from the concept of approximating a function near a given point using the \"best\" linear approximation.\nThis approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point a, and the slope of the line is the derivative of the function at a.\nA function is **differentiable at a** if the [limit](https://bliptext.com/articles/limit-of-a-function)\nexists.\nThis limit is known as the ****derivative** **o**f**** **f** at a**, and the **f**unction **f**', possibly de**f**ined on only a subset **o**f**** \\mathbb{R}, is the **derivative** (or **derivative** **f**unction) **o**f**** **f**.\nIf the derivative exists everywhere, the function is said to be **differentiable**.\nAs **a** simple consequence of the definition, f is continuous **a**t **a** if it is differenti**a**ble there.\nDifferentiability is therefore a stronger regularity condition (condition describing the \"smoothness\" of a function) than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere (see [Weierstrass's nowhere differentiable continuous function](https://bliptext.com/articles/weierstrass-function)).\nIt is possible to discuss the existence of higher-order derivatives as well, by finding the derivative of a derivative function, and so on.\nOne can classify functions by their **differentiability class**.\nThe class C^0 (sometimes C^0([a,b]) to indicate the interval of applicability) consists of all continuous functions.\nThe class C^1 consists of all [differentiable functions](https://bliptext.com/articles/differentiable-function) whose derivative is continuous; such functions are called **continuously differentiable**.\nThus, a C^1 function is exactly a function whose derivative exists and is of class C^0.\nIn general, the classes C^k can be defined [recursively](https://bliptext.com/articles/recursion) by declaring C^0 to be the set of all continuous functions and declaring C^k for any positive integer k to be the set of all differentiable functions whose derivative is in C^{k-1}.\nIn particular, C^k is contained in C^{k-1} for every k, and there are examples to show that this containment is strict.\nClass C^\\infty is the intersection of the sets C^k as k varies over the non-negative integers, and the members of this class are known as the **smooth functions**.\nClass C^\\omega consists of all [analytic functions](https://bliptext.com/articles/analytic-function), and is strictly contained in C^\\infty (see [bump function](https://bliptext.com/articles/bump-function) for a smooth function that is not analytic).\n\n## Series\nA series formalizes the imprecise notion of taking the sum of an endless sequence of numbers.\nThe idea that taking the sum of an \"infinite\" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers.\nThe modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an \"infinite\" number of terms.\nInstead, the finite sum of the first n terms of the sequence, known as a partial sum, is considered, and the concept of a limit is applied to the sequence of partial sums as n grows without bound.\nThe series is assigned the value of this limit, if it exists.\nGiven an (infinite) [sequence](https://bliptext.com/articles/sequence) (a_n), we can define an associated **series** as the formal mathematical object, sometimes simply written as \\sum a_n.\nThe **partial sums** of a series \\sum a_n are the numbers.\nA series \\sum a_n is said to be **convergent** if the sequence consisting of its partial sums, (s_n), is **convergent**; otherwise it is **divergent**.\nThe **sum** of a convergent series is defined as the number.\nThe word \"sum\" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply \"adding\" an infinite number of terms.\nFor instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the [Riemann rearrangement theorem](https://bliptext.com/articles/riemann-rearrangement-theorem) for further discussion).\nAn example of a convergent series is a [geometric series](https://bliptext.com/articles/geometric-series) which forms the basis of one of Zeno's famous [paradoxes](https://bliptext.com/articles/zeno-s-paradoxes):\nIn contrast, the [harmonic series](https://bliptext.com/articles/harmonic-series-mathematics) has been known since the Middle Ages to be a divergent series:\n(Here, \"=\\infty\" is merely a notational convention to indicate that the partial sums of the series grow without bound.)\nA series \\sum a_n is said to **[converge absolutely](https://bliptext.com/articles/absolute-convergence)** if \\sum |a_n| is convergent.\nA **converge**nt series \\sum a_n for which \\sum |a_n| diverges is said to **converge** **non-absolutely**.\nIt is easily shown that absolute convergence of a series implies its convergence.\nOn the other hand, an example of a series that converges non-absolutely is\n\n### Taylor series\nThe Taylor series of a [[real](https://bliptext.com/articles/real-number)](https://bliptext.com/articles/[real](https://bliptext.com/articles/real-number)-valued-function) or [complex-valued function](https://bliptext.com/articles/complex-valued-function) ƒ(x) that is [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable-function) at a [[real](https://bliptext.com/articles/real-number)](https://bliptext.com/articles/[real](https://bliptext.com/articles/real-number)-valued-function) or [complex number](https://bliptext.com/articles/complex-number) a is the [power series](https://bliptext.com/articles/power-series)\nwhich can be written in the more compact [sigma notation](https://bliptext.com/articles/summation) as\nwhere n!\ndenotes the [factorial](https://bliptext.com/articles/factorial) of n and ƒ(n)(a) denotes the nth [derivative](https://bliptext.com/articles/derivative) of ƒ evaluated at the point a.\nThe derivative of order zero ƒ is defined to be ƒ itself and (x − a)0 and 0!\nare both defined to be 1.\nIn the case that, the series is also called a Maclaurin series.\nA Taylor series of f about point a may diverge, converge at only the point a, converge for all x such that |x-a| [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) > [Fourier analysis](https://bliptext.com/articles/fourier-analysis).\n# [The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis. | upload.wikimedia.org/wikipedia/commons/2/2c/Fourier///Series.svg]\n\n## Integration\nIntegration is a formalization of the problem of finding the area bound by a curve and the related problems of determining the length of a curve or volume enclosed by a surface.\nThe basic strategy to solving problems of this type was known to the ancient Greeks and Chinese, and was known as the [method of exhaustion](https://bliptext.com/articles/method-of-exhaustion).\nGenerally speaking, the desired area is bounded from above and below, respectively, by increasingly accurate circumscribing and inscribing polygonal approximations whose exact areas can be computed.\nBy considering approximations consisting of a larger and larger (\"infinite\") number of smaller and smaller (\"infinitesimal\") pieces, the area bound by the curve can be deduced, as the upper and lower bounds defined by the approximations converge around a common value.\nThe spirit of this basic strategy can easily be seen in the definition of the Riemann integral, in which the integral is said to exist if upper and lower Riemann (or Darboux) sums converge to a common value as thinner and thinner rectangular slices (\"refinements\") are considered.\nThough the machinery used to define it is much more elaborate compared to the Riemann integral, the Lebesgue integral was defined with similar basic ideas in mind.\nCompared to the Riemann integral, the more sophisticated Lebesgue integral allows area (or length, volume, etc.; termed a \"measure\" in general) to be defined and computed for much more complicated and irregular subsets of Euclidean space, although there still exist \"non-measurable\" subsets for which an area cannot be assigned.\n\n### Riemann integration\nThe Riemann integral is defined in terms of [Riemann sums](https://bliptext.com/articles/riemann-sum) of functions with respect to tagged partitions of an interval.\nLet [a,b] be a [closed interval](https://bliptext.com/articles/interval-mathematics) of the real line; then a **tagged partition** \\cal{P} of [a,b] is a finite sequence\nThis partitions the interval [a,b] into n sub-intervals indexed by, each of which is \"tagged\" with a distinguished point.\nFor a function f bounded on [a,b], we define the **Riemann sum** of f with respect to tagged partition \\cal{P} as\nwhere is the width of sub-interval i. Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width.\nThe **mesh** of such a tagged partition is the width of the largest sub-interval formed by the partition,.\nWe say that the **Riemann integral** of f on [a,b] is S if for any there exists \\delta>0 such that, for any tagged partition \\cal{P} with mesh, we have\nThis is sometimes denoted.\nWhen the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) **Darboux sum**.\nA function is **Darboux integrable** if the upper and lower [Darboux sums](https://bliptext.com/articles/darboux-integral) can be made to be arbitrarily close to each other for a sufficiently small mesh.\nAlthough this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal.\nIn fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former.\nThe [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) asserts that integration and differentiation are inverse operations in a certain sense.\n\n### Lebesgue integration and measure\n**Lebesgue integration** is a mathematical construction that extends the integral to a larger class of functions; it also extends the [domains](https://bliptext.com/articles/domain-of-a-function) on which these functions can be defined.\nThe concept of a **measure**, an abstraction of length, area, or volume, is central to Lebesgue integral [probability theory](https://bliptext.com/articles/probability-theory).\n\n## Distributions\n**Distributions** (or **[generalized [functions](https://bliptext.com/articles/function-mathematics)](https://bliptext.com/articles/generalized-[functions](https://bliptext.com/articles/function-mathematics))**) are objects that generalize [functions](https://bliptext.com/articles/function-mathematics).\nDistributions make it possible to [differentiate](https://bliptext.com/articles/derivative) functions whose derivatives do not exist in the classical sense.\nIn particular, any [locally integrable](https://bliptext.com/articles/locally-integrable) function has a distributional derivative.\n\n## Relation to complex analysis\nReal [analysis](https://bliptext.com/articles/mathematical-analysis) is an area of [analysis](https://bliptext.com/articles/mathematical-analysis) that studies concepts such as sequences and their limits, continuity, [differentiation](https://bliptext.com/articles/derivative), [integration](https://bliptext.com/articles/integral) and sequences of functions.\nBy definition, real analysis focuses on the [real numbers](https://bliptext.com/articles/real-number), often including positive and negative [infinity](https://bliptext.com/articles/infinity-mathematics) to form the [extended real line](https://bliptext.com/articles/extended-real-line).\nReal analysis is closely related to [complex analysis](https://bliptext.com/articles/complex-analysis), which studies broadly the same properties of [complex numbers](https://bliptext.com/articles/complex-number).\nIn complex analysis, it is natural to define [differentiation](https://bliptext.com/articles/derivative) via [holomorphic functions](https://bliptext.com/articles/holomorphic-functions), which have a number of useful properties, such as repeated differentiability, expressibility as [power series](https://bliptext.com/articles/power-series), and satisfying the [Cauchy integral formula](https://bliptext.com/articles/cauchy-integral-formula).\nIn real analysis, it is usually more natural to consider [differentiable](https://bliptext.com/articles/differentiable-function), [smooth](https://bliptext.com/articles/smooth-functions), or [harmonic functions](https://bliptext.com/articles/harmonic-functions), which are more widely applicable, but may lack some more powerful properties of holomorphic functions.\nHowever, results such as the [fundamental theorem of algebra](https://bliptext.com/articles/fundamental-theorem-of-algebra) are simpler when expressed in terms of complex numbers.\nTechniques from the [theory of analytic functions](https://bliptext.com/articles/theory-of-analytic-functions) of a complex variable are often used in real analysis – such as evaluation of real integrals by [residue calculus](https://bliptext.com/articles/residue-theorem).\n\n# Important results\nImportant results include the [Bolzano–Weierstrass](https://bliptext.com/articles/bolzano-weierstrass-theorem) and [Heine–Borel theorems](https://bliptext.com/articles/heine-borel-theorem), the [intermediate value theorem](https://bliptext.com/articles/intermediate-value-theorem) and [mean value theorem](https://bliptext.com/articles/mean-value-theorem), [Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem), the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus), the [Arzelà-Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem), the [Stone-Weierstrass theorem](https://bliptext.com/articles/stone-weierstrass-theorem), [Fatou's lemma](https://bliptext.com/articles/fatou-s-lemma), and the [monotone convergence](https://bliptext.com/articles/monotone-convergence-theorem) and [dominated convergence theorems](https://bliptext.com/articles/dominated-convergence-theorem).\n\n# Generalizations and related areas of mathematics\nVarious ideas from real analysis can be generalized from the real line to broader or more abstract contexts.\nThese generalizations link real analysis to other disciplines and subdisciplines.\nFor instance, generalization of ideas like continuous functions and compactness from real analysis to [metric spaces](https://bliptext.com/articles/metric-space) and [topological spaces](https://bliptext.com/articles/topological-space) connects real analysis to the field of [general topology](https://bliptext.com/articles/general-topology), while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of [Banach spaces](https://bliptext.com/articles/banach-space) and [Hilbert spaces](https://bliptext.com/articles/hilbert-space) and, more generally to [functional analysis](https://bliptext.com/articles/functional-analysis).\n[Georg Cantor](https://bliptext.com/articles/georg-cantor)'s investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth to [naive set theory](https://bliptext.com/articles/naive-set-theory).\nThe study of issues of [convergence](https://bliptext.com/articles/limit-mathematics) for sequences of functions eventually gave rise to [Fourier analysis](https://bliptext.com/articles/fourier-analysis) as a subdiscipline of mathematical analysis.\nInvestigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of [holomorphic functions](https://bliptext.com/articles/holomorphic-function) and the inception of [complex analysis](https://bliptext.com/articles/complex-analysis) as another distinct subdiscipline of analysis.\nOn the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstract [measure spaces](https://bliptext.com/articles/measure-space), a fundamental concept in [measure theory](https://bliptext.com/articles/measure-theory).\nFinally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of [vector calculus](https://bliptext.com/articles/vector-calculus), whose further generalization and formalization played an important role in the evolution of the concepts of [differential forms](https://bliptext.com/articles/differential-form) and [smooth (differentiable) manifolds](https://bliptext.com/articles/differentiable-manifold) in [differential [geometry](https://bliptext.com/articles/geometry)](https://bliptext.com/articles/differential-[geometry](https://bliptext.com/articles/geometry)) and other closely related areas of [geometry](https://bliptext.com/articles/geometry) and [topology](https://bliptext.com/articles/topology).\n\n# Sources","revisionCount":3},"hasMore":false,"nextCursor":null}

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