{"history":[],"article":{"id":"c36e5de0-0ca6-4b55-9dc8-98b221ce4969","title":"Intermediate value theorem","slug":"intermediate-value-theorem","content":":::summary\n:::\n\n# \nIn [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), the **intermediate value theorem** states that if f is a [continuous](https://bliptext.com/articles/continuous-[function](https://bliptext.com/articles/function-mathematics)) [function](https://bliptext.com/articles/function-mathematics) whose [domain](https://bliptext.com/articles/domain-of-a-function) contains the [interval](https://bliptext.com/articles/interval-mathematics), then it takes on any given value between f(a) and f(b) at some point within the [interval](https://bliptext.com/articles/interval-mathematics).\nThis has two important [corollaries](https://bliptext.com/articles/corollary):\n# [Intermediate value theorem: Let f be a continuous function defined on [a,b] and let s be a number with . Then there exists some x between a and b such that f(x) = s. | upload.wikimedia.org/wikipedia/commons/8/83/Illustration///for///the///intermediate///value///theorem.svg]\n\n# Motivation\nThis captures an intuitive property of continuous functions over the [real numbers](https://bliptext.com/articles/real-number): given f continuous on [1,2] with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2.\nIt represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper.\n# [The intermediate value theorem | upload.wikimedia.org/wikipedia/commons/e/e2/Intermediatevaluetheorem.svg]\n\n# Theorem\nThe intermediate value theorem states the following:\nConsider an interval I = [a,b] of real numbers \\R and a continuous function.\nThen\n**Remark:** Version II states that the [set](https://bliptext.com/articles/set-mathematics) of function values has no gap.\nFor any two function values with c < d all points in the interval are also function values,\nA subset of the real numbers with no internal gap is an interval.\nVersion I is naturally contained in Version II.\n\n# Relation to completeness\nThe theorem depends on, and is equivalent to, the [completeness of the real numbers](https://bliptext.com/articles/completeness-of-the-real-numbers).\nThe intermediate value theorem does not apply to the [rational numbers](https://bliptext.com/articles/rational-number) **Q** because gaps exist between [rational numbers](https://bliptext.com/articles/rational-number); ir[rational numbers](https://bliptext.com/articles/rational-number) fill those gaps.\nFor example, the function f(x) = x^2 for x\\in\\Q satisfies f(0) = 0 and f(2) = 4.\nHowever, there is no rational number x such that f(x)=2, because \\sqrt 2 is an irrational number.\n\n# Proof\n\n## Proof version A\nThe theorem may be proven as a consequence of the [completeness](https://bliptext.com/articles/completeness-order-theory) property of the real numbers as follows:\nWe shall prove the first case,.\nThe second case is similar.\nLet S be the set of all x \\in [a,b] such that f(x)u.\nHence c cannot be b.\nWith c \\neq a and c \\neq b, it must be the case c \\in (a,b).\nNow we claim that f(c)=u.\nFix some.\nSince f is continuous at c, such that,.\nSince c \\in (a,b) and (a,b) is open, such that.\nSet.\nThen we have\nfor all.\nBy the properties of the supremum, there exists some that is contained in S, and so\nPicking, we know that because c is the supremum of S. This means that\nBoth inequalities\nare valid for all, from which we deduce f(c) = u as the only possible value, as stated.\n\n## Proof version B\nWe will only prove the case of f(a)u>f(b) case is similar.\nDefine g(x)=f(x)-u which is equivalent to f(x)=g(x)+u and lets us rewrite f(a)0 such that implies, that , which is equivalent to g(x)<0.\nIf we just chose, where , then as 1 < N, xc, contradicting the **upper bound** property of the least **upper bound** c, so g(c)\\geq 0.\nAssume then, that g(c)>0.\nWe similarly chose and know, that there exists a \\delta>0 such that implies.\nWe can rewrite this as which implies, that g(x)>0.\nIf we now chose, then g(x)>0 and a0 is impossible.\nIf we combine both results, we get that g(c)=0 or f(c)=u is the only remaining possibility.\n**Remark:** The intermediate value theorem can also be proved using the methods of [non-standard analysis](https://bliptext.com/articles/non-standard-analysis), which places \"intuitive\" arguments involving infinitesimals on a rigorous footing.\n\n# History\nA form of the theorem was postulated as early as the 5th century BCE, in the work of [Bryson of Heraclea](https://bliptext.com/articles/bryson-of-heraclea) on [squaring the circle](https://bliptext.com/articles/squaring-the-circle).\nBryson argued that, as circles larger than and smaller than a given square both exist, there must exist a circle of equal area.\nThe theorem was first proved by [Bernard Bolzano](https://bliptext.com/articles/bernard-bolzano) in 1817.\nBolzano used the following formulation of the theorem:\nLet f, \\varphi be continuous functions on the interval between \\alpha and \\beta such that and.\nThen there is an x between \\alpha and \\beta such that.\nThe equivalence between this formulation and the modern one can be shown by setting \\varphi to the appropriate constant function.\n[Augustin-Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) provided the modern formulation and a proof in 1821.\nBoth were inspired by the goal of formalizing the analysis of functions and the work of [Joseph-Louis Lagrange](https://bliptext.com/articles/joseph-louis-lagrange).\nThe idea that continuous functions possess the intermediate value property has an earlier origin.\n[Simon Stevin](https://bliptext.com/articles/simon-stevin) proved the intermediate value theorem for [polynomials](https://bliptext.com/articles/polynomial) (using a [cubic](https://bliptext.com/articles/cubic-function) as an example) by providing an algorithm for constructing the decimal expansion of the solution.\nThe algorithm iteratively subdivides the interval into 10 parts, producing an additional decimal digit at each step of the iteration.\nBefore the formal definition of continuity was given, the intermediate value property was given as part of the definition of a continuous function.\nProponents include [Louis Arbogast](https://bliptext.com/articles/louis-arbogast), who assumed the functions to have no jumps, satisfy the intermediate value property and have increments whose sizes corresponded to the sizes of the increments of the variable.\nEarlier authors held the result to be intuitively obvious and requiring no proof.\nThe insight of Bolzano and Cauchy was to define a general notion of continuity (in terms of [infinitesimals](https://bliptext.com/articles/infinitesimal) in Cauchy's case and using real inequalities in Bolzano's case), and to provide a proof based on such definitions.\n\n# Converse is false\nA [Darboux function](https://bliptext.com/articles/darboux-function) is a real-valued function f that has the \"intermediate value property,\" i.e., that satisfies the conclusion of the intermediate value theorem: for any two values a and b in the domain of f, and any y between\nf(a)\nand\nf(b)\n, there is some c between a and b with\nf(c) = y\n. The intermediate value theorem says that every continuous function is a Darboux function.\nHowever, not every Darboux function is continuous; i.e., the converse of the intermediate value theorem is false.\nAs an example, take the function\nf : [0, ∞) → [−1, 1]\ndefined by\nf(x) = sin(1/x)\nfor\nx > 0\nand\nf(0) = 0\n. This function is not continuous at\nx = 0\nbecause the [limit](https://bliptext.com/articles/limit-of-a-function) of\nf(x)\nas x tends to 0 does not exist; yet the function has the intermediate value property.\nAnother, more complicated example is given by the [Conway base 13 function](https://bliptext.com/articles/conway-base-13-function).\nIn fact, [Darboux's theorem](https://bliptext.com/articles/darboux-s-theorem-analysis) states that all functions that result from the [differentiation](https://bliptext.com/articles/derivative) of some other function on some interval have the [intermediate value property](https://bliptext.com/articles/intermediate-value-property) (even though they need not be continuous).\nHistorically, this intermediate value property has been suggested as a definition for continuity of real-valued functions; this definition was not adopted.\n\n# Generalizations\n\n## Multi-dimensional spaces\nThe [Poincaré-Miranda theorem](https://bliptext.com/articles/poincar-miranda-theorem) is a generalization of the Intermediate value theorem from a (one-dimensional) interval to a (two-dimensional) rectangle, or more generally, to an n-dimensional [cube](https://bliptext.com/articles/n-cube).\nVrahatis presents a similar generalization to triangles, or more generally, n-dimensional [simplices](https://bliptext.com/articles/simplex).\nLet Dn be an n-dimensional simplex with n+1 vertices denoted by v0,...,vn.\nLet F=(f1,...,fn) be a continuous function from Dn to Rn, that never equals 0 on the boundary of Dn.\nSuppose F satisfies the following conditions:\nThen there is a point z in the [interior](https://bliptext.com/articles/interior-topology) of Dn on which F(z)=(0,...,0).\nIt is possible to normalize the fi such that fi(vi)>0 for all i; then the conditions become simpler:\nThe theorem can be proved based on the [Knaster–Kuratowski–Mazurkiewicz lemma](https://bliptext.com/articles/knaster-kuratowski-mazurkiewicz-lemma).\nIn can be used for approximations of fixed points and zeros.\n\n## General metric and topological spaces\nThe intermediate value theorem is closely linked to the [topological](https://bliptext.com/articles/topology) notion of [connectedness](https://bliptext.com/articles/connectedness-topology) and follows from the basic properties of connected sets in metric spaces and connected subsets of **R** in particular:\nIn fact, connectedness is a [topological property](https://bliptext.com/articles/topological-property) and generalizes to [topological spaces](https://bliptext.com/articles/topological-space): If X and Y are [topological spaces](https://bliptext.com/articles/topological-space), is a continuous map, and X is a [connected space](https://bliptext.com/articles/connected-space), then f(X) is connected. The preservation of connectedness under continuous maps can be thought of as a generalization of the intermediate value theorem, a property of continuous, real-valued functions of a real variable, to continuous functions in general spaces.\nRecall the first version of the intermediate value theorem, stated previously:\nThe intermediate value theorem is an immediate consequence of these two properties of connectedness:\nThe intermediate value theorem generalizes in a natural way: Suppose that X is a connected topological space and\n(Y, <)\nis a [totally ordered](https://bliptext.com/articles/total-order) set equipped with the [order topology](https://bliptext.com/articles/order-topology), and let\nf : X → Y\nbe a continuous map.\nIf a and b are two points in X and u is a point in Y lying between\nf(a)\nand\nf(b)\nwith respect to\n<\n, then there exists c in X such that\nf(c) = u\n. The original theorem is recovered by noting that\nR\nis connected and that its natural [topology](https://bliptext.com/articles/topological-space) is the order [topology](https://bliptext.com/articles/topological-space).\nThe [Brouwer fixed-point theorem](https://bliptext.com/articles/brouwer-fixed-point-theorem) is a related theorem that, in one dimension, gives a special case of the intermediate value theorem.\n\n# In constructive mathematics\nIn [constructive mathematics](https://bliptext.com/articles/constructive-mathematics), the intermediate value theorem is not true.\nInstead, one has to weaken the conclusion:\n\n# Practical applications\nA similar result is the [Borsuk–Ulam theorem](https://bliptext.com/articles/borsuk-ulam-theorem), which says that a continuous map from the n-sphere to Euclidean n-space will always map some pair of antipodal points to the same place.\nIn general, for any continuous function whose domain is some closed convex n-dimensional shape and any point inside the shape (not necessarily its center), there exist two antipodal points with respect to the given point whose functional value is the same.\nThe theorem also underpins the explanation of why rotating a wobbly table will bring it to stability (subject to certain easily met constraints).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:21:23.387Z","updated_at":"2025-01-25T17:21:23.387Z","search_vector":"'intermedi':1 'theorem':3 'valu':2","search_content":" \n \n\n# \nIn [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), the **intermediate value theorem** states that if f is a [continuous](https://bliptext.com/articles/continuous-[function](https://bliptext.com/articles/function-mathematics)) [function](https://bliptext.com/articles/function-mathematics) whose [domain](https://bliptext.com/articles/domain-of-a-function) contains the [interval](https://bliptext.com/articles/interval-mathematics), then it takes on any given value between f(a) and f(b) at some point within the [interval](https://bliptext.com/articles/interval-mathematics).\nThis has two important [corollaries](https://bliptext.com/articles/corollary):\n# [Intermediate value theorem: Let f be a continuous function defined on [a,b] and let s be a number with . Then there exists some x between a and b such that f(x) = s. | upload.wikimedia.org/wikipedia/commons/8/83/Illustration///for///the///intermediate///value///theorem.svg]\n\n# Motivation\nThis captures an intuitive property of continuous functions over the [real numbers](https://bliptext.com/articles/real-number): given f continuous on [1,2] with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2.\nIt represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper.\n# [The intermediate value theorem | upload.wikimedia.org/wikipedia/commons/e/e2/Intermediatevaluetheorem.svg]\n\n# Theorem\nThe intermediate value theorem states the following:\nConsider an interval I = [a,b] of real numbers \\R and a continuous function.\nThen\n**Remark:** Version II states that the [set](https://bliptext.com/articles/set-mathematics) of function values has no gap.\nFor any two function values with c u.\nHence c cannot be b.\nWith c \\neq a and c \\neq b, it must be the case c \\in (a,b).\nNow we claim that f(c)=u.\nFix some.\nSince f is continuous at c, such that,.\nSince c \\in (a,b) and (a,b) is open, such that.\nSet.\nThen we have\nfor all.\nBy the properties of the supremum, there exists some that is contained in S, and so\nPicking, we know that because c is the supremum of S. This means that\nBoth inequalities\nare valid for all, from which we deduce f(c) = u as the only possible value, as stated.\n\n## Proof version B\nWe will only prove the case of f(a) u>f(b) case is similar.\nDefine g(x)=f(x)-u which is equivalent to f(x)=g(x)+u and lets us rewrite f(a) 0 such that implies, that , which is equivalent to g(x) c, contradicting the **upper bound** property of the least **upper bound** c, so g(c)\\geq 0.\nAssume then, that g(c)>0.\nWe similarly chose and know, that there exists a \\delta>0 such that implies.\nWe can rewrite this as which implies, that g(x)>0.\nIf we now chose, then g(x)>0 and a 0 is impossible.\nIf we combine both results, we get that g(c)=0 or f(c)=u is the only remaining possibility.\n**Remark:** The intermediate value theorem can also be proved using the methods of [non-standard analysis](https://bliptext.com/articles/non-standard-analysis), which places \"intuitive\" arguments involving infinitesimals on a rigorous footing.\n\n# History\nA form of the theorem was postulated as early as the 5th century BCE, in the work of [Bryson of Heraclea](https://bliptext.com/articles/bryson-of-heraclea) on [squaring the circle](https://bliptext.com/articles/squaring-the-circle).\nBryson argued that, as circles larger than and smaller than a given square both exist, there must exist a circle of equal area.\nThe theorem was first proved by [Bernard Bolzano](https://bliptext.com/articles/bernard-bolzano) in 1817.\nBolzano used the following formulation of the theorem:\nLet f, \\varphi be continuous functions on the interval between \\alpha and \\beta such that and.\nThen there is an x between \\alpha and \\beta such that.\nThe equivalence between this formulation and the modern one can be shown by setting \\varphi to the appropriate constant function.\n[Augustin-Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) provided the modern formulation and a proof in 1821.\nBoth were inspired by the goal of formalizing the analysis of functions and the work of [Joseph-Louis Lagrange](https://bliptext.com/articles/joseph-louis-lagrange).\nThe idea that continuous functions possess the intermediate value property has an earlier origin.\n[Simon Stevin](https://bliptext.com/articles/simon-stevin) proved the intermediate value theorem for [polynomials](https://bliptext.com/articles/polynomial) (using a [cubic](https://bliptext.com/articles/cubic-function) as an example) by providing an algorithm for constructing the decimal expansion of the solution.\nThe algorithm iteratively subdivides the interval into 10 parts, producing an additional decimal digit at each step of the iteration.\nBefore the formal definition of continuity was given, the intermediate value property was given as part of the definition of a continuous function.\nProponents include [Louis Arbogast](https://bliptext.com/articles/louis-arbogast), who assumed the functions to have no jumps, satisfy the intermediate value property and have increments whose sizes corresponded to the sizes of the increments of the variable.\nEarlier authors held the result to be intuitively obvious and requiring no proof.\nThe insight of Bolzano and Cauchy was to define a general notion of continuity (in terms of [infinitesimals](https://bliptext.com/articles/infinitesimal) in Cauchy's case and using real inequalities in Bolzano's case), and to provide a proof based on such definitions.\n\n# Converse is false\nA [Darboux function](https://bliptext.com/articles/darboux-function) is a real-valued function f that has the \"intermediate value property,\" i.e., that satisfies the conclusion of the intermediate value theorem: for any two values a and b in the domain of f, and any y between\nf(a)\nand\nf(b)\n, there is some c between a and b with\nf(c) = y\n. The intermediate value theorem says that every continuous function is a Darboux function.\nHowever, not every Darboux function is continuous; i.e., the converse of the intermediate value theorem is false.\nAs an example, take the function\nf : [0, ∞) → [−1, 1]\ndefined by\nf(x) = sin(1/x)\nfor\nx > 0\nand\nf(0) = 0\n. This function is not continuous at\nx = 0\nbecause the [limit](https://bliptext.com/articles/limit-of-a-function) of\nf(x)\nas x tends to 0 does not exist; yet the function has the intermediate value property.\nAnother, more complicated example is given by the [Conway base 13 function](https://bliptext.com/articles/conway-base-13-function).\nIn fact, [Darboux's theorem](https://bliptext.com/articles/darboux-s-theorem-analysis) states that all functions that result from the [differentiation](https://bliptext.com/articles/derivative) of some other function on some interval have the [intermediate value property](https://bliptext.com/articles/intermediate-value-property) (even though they need not be continuous).\nHistorically, this intermediate value property has been suggested as a definition for continuity of real-valued functions; this definition was not adopted.\n\n# Generalizations\n\n## Multi-dimensional spaces\nThe [Poincaré-Miranda theorem](https://bliptext.com/articles/poincar-miranda-theorem) is a generalization of the Intermediate value theorem from a (one-dimensional) interval to a (two-dimensional) rectangle, or more generally, to an n-dimensional [cube](https://bliptext.com/articles/n-cube).\nVrahatis presents a similar generalization to triangles, or more generally, n-dimensional [simplices](https://bliptext.com/articles/simplex).\nLet Dn be an n-dimensional simplex with n+1 vertices denoted by v0,...,vn.\nLet F=(f1,...,fn) be a continuous function from Dn to Rn, that never equals 0 on the boundary of Dn.\nSuppose F satisfies the following conditions:\nThen there is a point z in the [interior](https://bliptext.com/articles/interior-topology) of Dn on which F(z)=(0,...,0).\nIt is possible to normalize the fi such that fi(vi)>0 for all i; then the conditions become simpler:\nThe theorem can be proved based on the [Knaster–Kuratowski–Mazurkiewicz lemma](https://bliptext.com/articles/knaster-kuratowski-mazurkiewicz-lemma).\nIn can be used for approximations of fixed points and zeros.\n\n## General metric and topological spaces\nThe intermediate value theorem is closely linked to the [topological](https://bliptext.com/articles/topology) notion of [connectedness](https://bliptext.com/articles/connectedness-topology) and follows from the basic properties of connected sets in metric spaces and connected subsets of **R** in particular:\nIn fact, connectedness is a [topological property](https://bliptext.com/articles/topological-property) and generalizes to [topological spaces](https://bliptext.com/articles/topological-space): If X and Y are [topological spaces](https://bliptext.com/articles/topological-space), is a continuous map, and X is a [connected space](https://bliptext.com/articles/connected-space), then f(X) is connected. The preservation of connectedness under continuous maps can be thought of as a generalization of the intermediate value theorem, a property of continuous, real-valued functions of a real variable, to continuous functions in general spaces.\nRecall the first version of the intermediate value theorem, stated previously:\nThe intermediate value theorem is an immediate consequence of these two properties of connectedness:\nThe intermediate value theorem generalizes in a natural way: Suppose that X is a connected topological space and\n(Y, <)\nis a [totally ordered](https://bliptext.com/articles/total-order) set equipped with the [order topology](https://bliptext.com/articles/order-topology), and let\nf : X → Y\nbe a continuous map.\nIf a and b are two points in X and u is a point in Y lying between\nf(a)\nand\nf(b)\nwith respect to\n<\n, then there exists c in X such that\nf(c) = u\n. The original theorem is recovered by noting that\nR\nis connected and that its natural [topology](https://bliptext.com/articles/topological-space) is the order [topology](https://bliptext.com/articles/topological-space).\nThe [Brouwer fixed-point theorem](https://bliptext.com/articles/brouwer-fixed-point-theorem) is a related theorem that, in one dimension, gives a special case of the intermediate value theorem.\n\n# In constructive mathematics\nIn [constructive mathematics](https://bliptext.com/articles/constructive-mathematics), the intermediate value theorem is not true.\nInstead, one has to weaken the conclusion:\n\n# Practical applications\nA similar result is the [Borsuk–Ulam theorem](https://bliptext.com/articles/borsuk-ulam-theorem), which says that a continuous map from the n-sphere to Euclidean n-space will always map some pair of antipodal points to the same place.\nIn general, for any continuous function whose domain is some closed convex n-dimensional shape and any point inside the shape (not necessarily its center), there exist two antipodal points with respect to the given point whose functional value is the same.\nThe theorem also underpins the explanation of why rotating a wobbly table will bring it to stability (subject to certain easily met constraints).","revisionCount":0},"hasMore":false,"nextCursor":null}