{"history":[],"article":{"id":"d7566ae8-2f40-454e-b24d-c5071bd8ac03","title":"Equicontinuity","slug":"equicontinuity","content":":::summary\n:::\n\n# \nIn [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), a family of functions is **equi[continuous](https://bliptext.com/articles/continuous-function)** if all the functions are [continuous](https://bliptext.com/articles/continuous-function) and they have equal variation over a given [neighbourhood](https://bliptext.com/articles/neighbourhood-mathematics), in a precise sense described herein.\nIn particular, the concept applies to [countable](https://bliptext.com/articles/countable-set) families, and thus sequences of functions.\nEquicontinuity appears in the formulation of [Ascoli's theorem](https://bliptext.com/articles/ascoli-s-theorem), which states that a subset of C(X), the space of [continuous functions on a compact Hausdorff space](https://bliptext.com/articles/continuous-functions-on-a-compact-hausdorff-space) X, is compact if and only if it is closed, pointwise bounded and equicontinuous.\nAs a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori).\nIn particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or [locally compact space](https://bliptext.com/articles/locally-compact-space) is continuous.\nIf, in addition, fn are [holomorphic](https://bliptext.com/articles/holomorphic), then the limit is also [holomorphic](https://bliptext.com/articles/holomorphic).\nThe uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.\n\n# Equicontinuity between metric spaces\nLet X and Y be two [metric spaces](https://bliptext.com/articles/metric-space), and F a family of functions from X to Y.\nWe shall denote by d the respective metrics of these spaces.\nThe family F is **equicontinuous at a point** x0 ∈ X if for every ε > 0, there exists a δ > 0 such that d(ƒ(x0), ƒ(x)) < ε for all ƒ ∈ F and all x such that d(x0, x) < δ.\nThe family is **pointwise equicontinuous** if it is equicontinuous at each point of X.\nThe family F is **uniformly equicontinuous** if for every ε > 0, there exists a δ > 0 such that d(ƒ(x1), ƒ(x2)) < ε for all ƒ ∈ F and all x1, x2 ∈ X such that d(x1, x2) < δ.\nFor comparison, the statement 'all functions ƒ in F are continuous' means that for every ε > 0, every ƒ ∈ F, and every x0 ∈ X, there exists a δ > 0 such that d(ƒ(x0), ƒ(x)) < ε for all x ∈ X such that d(x0, x) < δ.\nMore generally, when X is a topological space, a set F of functions from X to Y is said to be equicontinuous at x if for every ε > 0, x has a neighborhood Ux such that\nfor all y ∈ Ux and ƒ ∈ F.\nThis definition usually appears in the context of [topological vector spaces](https://bliptext.com/articles/topological-vector-space).\nWhen X is compact, a set is uniformly equicontinuous if and only if it is equicontinuous at every point, for essentially the same reason as that uniform continuity and continuity coincide on compact spaces.\nUsed on its own, the term \"equicontinuity\" may refer to either the pointwise or uniform notion, depending on the context.\nOn a compact space, these notions coincide.\nSome basic properties follow immediately from the definition.\nEvery finite set of continuous functions is equicontinuous.\nThe closure of an equicontinuous set is again equicontinuous.\nEvery member of a uniformly equicontinuous set of functions is [uniformly continuous](https://bliptext.com/articles/uniformly-continuous), and every finite set of [uniformly continuous](https://bliptext.com/articles/uniformly-continuous) functions is uniformly equicontinuous.\n\n## Examples\n\n## Counterexamples\n\n# Equicontinuity of maps valued in topological groups\nSuppose that T is a topological space and Y is an additive [topological group](https://bliptext.com/articles/topological-[group](https://bliptext.com/articles/group-algebra)) (i.e. a [group](https://bliptext.com/articles/group-algebra) endowed with a topology making its operations continuous).\n[Topological vector spaces](https://bliptext.com/articles/topological-vector-space) are prominent examples of topological groups and every topological group has an associated canonical [uniformity](https://bliptext.com/articles/uniform-space).\nt ∈ T\nif for every neighborhood V of 0 in Y, there exists some neighborhood U of t in T such that\nh(U) ⊆ h(t) + V\nfor every\nh ∈ H\n. We say that H is **equicontinuous** if it is **equicontinuous** at every point of T.\nNote that if H is equicontinuous at a point then every map in H is continuous at the point.\nClearly, every finite set of continuous maps from T into Y is equicontinuous.\n\n# Equicontinuous linear maps\nBecause every [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) is a topological group so the definition of an equicontinuous family of maps given for topological groups transfers to TVSs without change.\n\n## Characterization of equicontinuous linear maps\nA family H of maps of the form X \\to Y between two topological vector spaces is said to be x \\in X if for every neighborhood V of the origin in Y there exists some neighborhood U of the origin in X such that for all h \\in H.\nIf H is a family of maps and U is a set then let With notation, if U and V are sets then for all h \\in H if and only if\nLet X and Y be [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (TVSs) and H be a family of linear operators from X into Y. Then the following are equivalent:\n

H is equicontinuous;
H is equicontinuous at every point of X.
H is equicontinuous at some point of X.
H is equicontinuous at the origin.\n
for every neighborhood V of the origin in Y, is a neighborhood of the origin in X.
the closure of H in is equicontinuous.\n
the [balanced hull](https://bliptext.com/articles/balanced-set) of H is equicontinuous.

\nwhile if Y is [locally convex](https://bliptext.com/articles/locally-convex) then this list may be extended to include:\n

the [convex hull](https://bliptext.com/articles/convex-hull) of H is equicontinuous.

the [convex balanced hull](https://bliptext.com/articles/absolutely-convex-set) of H is equicontinuous.

\n\nwhile if X and Y are [locally convex](https://bliptext.com/articles/locally-convex) then this list may be extended to include:\n

for every continuous [seminorm](https://bliptext.com/articles/seminorm) q on Y, there exists a continuous [seminorm](https://bliptext.com/articles/seminorm) p on X such that for all h \\in H. \nwhile if X is [barreled](https://bliptext.com/articles/barreled-space) and Y is locally convex then this list may be extended to include:\n

H is bounded in ;

H is bounded in L_b(X; Y).\n\nwhile if X and Y are [Banach spaces](https://bliptext.com/articles/banach-space) then this list may be extended to include:\n

(that is, H is uniformly bounded in the [operator norm](https://bliptext.com/articles/operator-norm)).

\n\n\n### Characterization of equicontinuous linear functionals\nLet X be a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) over the field \\mathbb{F} with [continuous dual space](https://bliptext.com/articles/continuous-dual-space) X^{\\prime}.\nA family H of linear functionals on X is said to be x \\in X if for every neighborhood V of the origin in \\mathbb{F} there exists some neighborhood U of the origin in X such that for all h \\in H.\nFor any subset the following are equivalent:\n

H is equicontinuous.
H is equicontinuous at the origin.
H is equicontinuous at some point of X.
H is contained in the [polar](https://bliptext.com/articles/polar-set) of some neighborhood of the origin in X
the [(pre)polar](https://bliptext.com/articles/polar-set) of H is a neighborhood of the origin in X.
the [weak* closure](https://bliptext.com/articles/weak-topology) of H in X^{\\prime} is equicontinuous.
the [balanced hull](https://bliptext.com/articles/balanced-set) of H is equicontinuous.
the [convex hull](https://bliptext.com/articles/convex-hull) of H is equicontinuous.
the [convex balanced hull](https://bliptext.com/articles/absolutely-convex-set) of H is equicontinuous.

\nwhile if X is [normed](https://bliptext.com/articles/normed-space) then this list may be extended to include:\n

H is a strongly bounded subset of X^{\\prime}.

\n\nwhile if X is a [barreled space](https://bliptext.com/articles/barreled-space) then this list may be extended to include:\n

H is [relatively compact](https://bliptext.com/articles/relatively-compact) in the [weak* topology](https://bliptext.com/articles/weak-topology) on X^{\\prime}.

H is [weak* bounded](https://bliptext.com/articles/weak-topology) (that is, H is bounded in X^{\\prime}).

H is bounded in the topology of bounded convergence (that is, H is bounded in X^{\\prime}).

\n\n\n## Properties of equicontinuous linear maps\nThe [uniform boundedness principle](https://bliptext.com/articles/uniform-boundedness-principle) (also known as the Banach–Steinhaus theorem) states that a set H of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that is, for each x \\in X. The result can be generalized to a case when Y is locally convex and X is a [barreled space](https://bliptext.com/articles/barreled-space).\n\n### Properties of equicontinuous linear functionals\n[Alaoglu's theorem](https://bliptext.com/articles/alaoglu-s-theorem) implies that the weak-* closure of an equicontinuous subset of X^{\\prime} is weak-* compact; thus that every equicontinuous subset is weak-* relatively compact.\nIf X is any locally convex TVS, then the family of all [barrels](https://bliptext.com/articles/barrelled-space) in X and the family of all subsets of X^{\\prime} that are convex, balanced, closed, and bounded in correspond to each other by polarity (with respect to ).\nIt follows that a locally convex TVS X is barreled if and only if every bounded subset of is equicontinuous.\n\n# Equicontinuity and uniform convergence\nLet X be a compact Hausdorff space, and equip C(X) with the [uniform norm](https://bliptext.com/articles/uniform-norm), thus making C(X) a [Banach space](https://bliptext.com/articles/banach-space), hence a metric space.\nThen [Arzelà–Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem) states that a subset of C(X) is compact if and only if it is closed, uniformly bounded and equicontinuous.\nThis is analogous to the [Heine–Borel theorem](https://bliptext.com/articles/heine-borel-theorem), which states that subsets of **R**n are compact if and only if they are closed and bounded.\nAs a corollary, every uniformly bounded equicontinuous sequence in C(X) contains a subsequence that converges uniformly to a continuous function on X.\nIn view of Arzelà–Ascoli theorem, a sequence in C(X) converges uniformly if and only if it is equicontinuous and converges pointwise.\nThe hypothesis of the statement can be weakened a bit: a sequence in C(X) converges uniformly if it is equicontinuous and converges pointwise on a dense subset to some function on X (not assumed continuous).\nThis weaker version is typically used to prove Arzelà–Ascoli theorem for separable compact spaces.\nAnother consequence is that the limit of an equicontinuous pointwise convergent sequence of continuous functions on a metric space, or on a locally compact space, is continuous.\n(See below for an example.)\nIn the above, the hypothesis of compactness of X cannot be relaxed.\nTo see that, consider a compactly supported continuous function g on ****R**** with g(0) = 1, and consider the equicontinuous sequence of functions on ****R**** defined by ƒn(x) = g(x − n).\nThen, ƒn converges pointwise to 0 but does not converge uniformly to 0.\nThis criterion for uniform convergence is often useful in real and complex analysis.\nSuppose we are given a sequence of continuous functions that converges pointwise on some open subset G of **R**n. As noted above, it actually converges uniformly on a compact subset of G if it is equicontinuous on the compact set.\nIn practice, showing the equicontinuity is often not so difficult.\nFor example, if the sequence consists of differentiable functions or functions with some regularity (e.g., the functions are solutions of a differential equation), then the [mean value theorem](https://bliptext.com/articles/mean-value-theorem) or some other kinds of estimates can be used to show the sequence is equicontinuous.\nIt then follows that the limit of the sequence is continuous on every compact subset of G; thus, continuous on G.\nA similar argument can be made when the functions are holomorphic.\nOne can use, for instance, [Cauchy's estimate](https://bliptext.com/articles/cauchy-s-estimate) to show the equicontinuity (on a compact subset) and conclude that the limit is holomorphic.\nNote that the equicontinuity is essential here.\nFor example, ƒn(x) = arctan n x converges to a multiple of the discontinuous [sign function](https://bliptext.com/articles/sign-function).\n\n# Generalizations\n\n## Equicontinuity in topological spaces\nThe most general scenario in which equicontinuity can be defined is for [topological spaces](https://bliptext.com/articles/topological-space) whereas uniform equicontinuity requires the [filter](https://bliptext.com/articles/filter-set-theory) of neighbourhoods of one point to be somehow comparable with the [filter](https://bliptext.com/articles/filter-set-theory) of neighbourhood of another point.\nThe latter is most generally done via a [uniform structure](https://bliptext.com/articles/uniform-structure), giving a [uniform space](https://bliptext.com/articles/uniform-space).\nAppropriate definitions in these cases are as follows:\nWe now briefly describe the basic idea underlying uniformities.\nThe uniformity 𝒱 is a non-empty collection of subsets of\nY × Y\nwhere, among many other properties, every\nV ∈ 𝒱\n, V contains the diagonal of Y (i.e.\n{(y, y) ∈ Y}\n). Every element of 𝒱 is called an **entourage**.\nUniformities generalize the idea (taken from [metric spaces](https://bliptext.com/articles/metric-spaces)) of points that are \"r-close\" (for\nr > 0\n), meaning that their distance is < r. To clarify this, suppose that\n(Y, d)\nis a metric space (so the diagonal of Y is the set\n{(y, z) ∈ Y × Y : d(y, z) = 0}\n) For any\nr > 0\n, let\ndenote the set of all pairs of points that are r-close.\nNote that if we were to \"forget\" that d existed then, for any\nr > 0\n, we would still be able to determine whether or not two points of Y are r-close by using only the sets\nUr\n. In this way, the sets\nUr\nencapsulate all the information necessary to define things such as [uniform continuity](https://bliptext.com/articles/uniform-continuity) and [uniform convergence](https://bliptext.com/articles/uniform-convergence) without needing any metric.\nAxiomatizing the most basic properties of these sets leads to the definition of a [uniformity](https://bliptext.com/articles/uniform-space).\nIndeed, the sets\nUr\ngenerate the uniformity that is canonically associated with the metric space\n(Y, d)\n.\nThe benefit of this generalization is that we may now extend some important definitions that make sense for metric spaces (e.g. [completeness](https://bliptext.com/articles/complete-metric-space)) to a broader category of topological spaces.\nIn particular, to [topological groups](https://bliptext.com/articles/topological-group) and [topological vector spaces](https://bliptext.com/articles/topological-vector-space).\n\n## Stochastic equicontinuity\nStochastic equicontinuity is a version of equicontinuity used in the context of sequences of functions of [random variables](https://bliptext.com/articles/random-variables), and their [convergence](https://bliptext.com/articles/convergence-of-random-variables).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:41:50.431Z","updated_at":"2025-01-25T19:41:50.431Z","search_vector":"'equicontinu':1","search_content":" \n \n\n# \nIn [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), a family of functions is **equi[continuous](https://bliptext.com/articles/continuous-function)** if all the functions are [continuous](https://bliptext.com/articles/continuous-function) and they have equal variation over a given [neighbourhood](https://bliptext.com/articles/neighbourhood-mathematics), in a precise sense described herein.\nIn particular, the concept applies to [countable](https://bliptext.com/articles/countable-set) families, and thus sequences of functions.\nEquicontinuity appears in the formulation of [Ascoli's theorem](https://bliptext.com/articles/ascoli-s-theorem), which states that a subset of C(X), the space of [continuous functions on a compact Hausdorff space](https://bliptext.com/articles/continuous-functions-on-a-compact-hausdorff-space) X, is compact if and only if it is closed, pointwise bounded and equicontinuous.\nAs a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori).\nIn particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or [locally compact space](https://bliptext.com/articles/locally-compact-space) is continuous.\nIf, in addition, fn are [holomorphic](https://bliptext.com/articles/holomorphic), then the limit is also [holomorphic](https://bliptext.com/articles/holomorphic).\nThe uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.\n\n# Equicontinuity between metric spaces\nLet X and Y be two [metric spaces](https://bliptext.com/articles/metric-space), and F a family of functions from X to Y.\nWe shall denote by d the respective metrics of these spaces.\nThe family F is **equicontinuous at a point** x0 ∈ X if for every ε > 0, there exists a δ > 0 such that d(ƒ(x0), ƒ(x)) 0, there exists a δ > 0 such that d(ƒ(x1), ƒ(x2)) 0, every ƒ ∈ F, and every x0 ∈ X, there exists a δ > 0 such that d(ƒ(x0), ƒ(x)) 0, x has a neighborhood Ux such that\nfor all y ∈ Ux and ƒ ∈ F.\nThis definition usually appears in the context of [topological vector spaces](https://bliptext.com/articles/topological-vector-space).\nWhen X is compact, a set is uniformly equicontinuous if and only if it is equicontinuous at every point, for essentially the same reason as that uniform continuity and continuity coincide on compact spaces.\nUsed on its own, the term \"equicontinuity\" may refer to either the pointwise or uniform notion, depending on the context.\nOn a compact space, these notions coincide.\nSome basic properties follow immediately from the definition.\nEvery finite set of continuous functions is equicontinuous.\nThe closure of an equicontinuous set is again equicontinuous.\nEvery member of a uniformly equicontinuous set of functions is [uniformly continuous](https://bliptext.com/articles/uniformly-continuous), and every finite set of [uniformly continuous](https://bliptext.com/articles/uniformly-continuous) functions is uniformly equicontinuous.\n\n## Examples\n\n## Counterexamples\n\n# Equicontinuity of maps valued in topological groups\nSuppose that T is a topological space and Y is an additive [topological [group](https://bliptext.com/articles/group-algebra)](https://bliptext.com/articles/topological-[group](https://bliptext.com/articles/group-algebra)) (i.e. a [group](https://bliptext.com/articles/group-algebra) endowed with a topology making its operations continuous).\n[Topological vector spaces](https://bliptext.com/articles/topological-vector-space) are prominent examples of topological groups and every topological group has an associated canonical [uniformity](https://bliptext.com/articles/uniform-space).\nt ∈ T\nif for every neighborhood V of 0 in Y, there exists some neighborhood U of t in T such that\nh(U) ⊆ h(t) + V\nfor every\nh ∈ H\n. We say that H is **equicontinuous** if it is **equicontinuous** at every point of T.\nNote that if H is equicontinuous at a point then every map in H is continuous at the point.\nClearly, every finite set of continuous maps from T into Y is equicontinuous.\n\n# Equicontinuous linear maps\nBecause every [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) is a topological group so the definition of an equicontinuous family of maps given for topological groups transfers to TVSs without change.\n\n## Characterization of equicontinuous linear maps\nA family H of maps of the form X \\to Y between two topological vector spaces is said to be x \\in X if for every neighborhood V of the origin in Y there exists some neighborhood U of the origin in X such that for all h \\in H.\nIf H is a family of maps and U is a set then let With notation, if U and V are sets then for all h \\in H if and only if\nLet X and Y be [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (TVSs) and H be a family of linear operators from X into Y. Then the following are equivalent:\n \n H is equicontinuous; \n H is equicontinuous at every point of X. \n H is equicontinuous at some point of X. \n H is equicontinuous at the origin.\n for every neighborhood V of the origin in Y, is a neighborhood of the origin in X. \n the closure of H in is equicontinuous.\n the [balanced hull](https://bliptext.com/articles/balanced-set) of H is equicontinuous. \n \nwhile if Y is [locally convex](https://bliptext.com/articles/locally-convex) then this list may be extended to include:\n the [convex hull](https://bliptext.com/articles/convex-hull) of H is equicontinuous. \n the [convex balanced hull](https://bliptext.com/articles/absolutely-convex-set) of H is equicontinuous. \n \nwhile if X and Y are [locally convex](https://bliptext.com/articles/locally-convex) then this list may be extended to include:\n for every continuous [seminorm](https://bliptext.com/articles/seminorm) q on Y, there exists a continuous [seminorm](https://bliptext.com/articles/seminorm) p on X such that for all h \\in H. \nwhile if X is [barreled](https://bliptext.com/articles/barreled-space) and Y is locally convex then this list may be extended to include:\n H is bounded in ; \n H is bounded in L_b(X; Y).\n \nwhile if X and Y are [Banach spaces](https://bliptext.com/articles/banach-space) then this list may be extended to include:\n (that is, H is uniformly bounded in the [operator norm](https://bliptext.com/articles/operator-norm)). \n \n\n### Characterization of equicontinuous linear functionals\nLet X be a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) over the field \\mathbb{F} with [continuous dual space](https://bliptext.com/articles/continuous-dual-space) X^{\\prime}.\nA family H of linear functionals on X is said to be x \\in X if for every neighborhood V of the origin in \\mathbb{F} there exists some neighborhood U of the origin in X such that for all h \\in H.\nFor any subset the following are equivalent:\n \n H is equicontinuous. \n H is equicontinuous at the origin. \n H is equicontinuous at some point of X. \n H is contained in the [polar](https://bliptext.com/articles/polar-set) of some neighborhood of the origin in X \n the [(pre)polar](https://bliptext.com/articles/polar-set) of H is a neighborhood of the origin in X. \n the [weak* closure](https://bliptext.com/articles/weak-topology) of H in X^{\\prime} is equicontinuous. \n the [balanced hull](https://bliptext.com/articles/balanced-set) of H is equicontinuous. \n the [convex hull](https://bliptext.com/articles/convex-hull) of H is equicontinuous. \n the [convex balanced hull](https://bliptext.com/articles/absolutely-convex-set) of H is equicontinuous. \n \nwhile if X is [normed](https://bliptext.com/articles/normed-space) then this list may be extended to include:\n H is a strongly bounded subset of X^{\\prime}. \n \nwhile if X is a [barreled space](https://bliptext.com/articles/barreled-space) then this list may be extended to include:\n H is [relatively compact](https://bliptext.com/articles/relatively-compact) in the [weak* topology](https://bliptext.com/articles/weak-topology) on X^{\\prime}. \n H is [weak* bounded](https://bliptext.com/articles/weak-topology) (that is, H is bounded in X^{\\prime}). \n H is bounded in the topology of bounded convergence (that is, H is bounded in X^{\\prime}). \n \n\n## Properties of equicontinuous linear maps\nThe [uniform boundedness principle](https://bliptext.com/articles/uniform-boundedness-principle) (also known as the Banach–Steinhaus theorem) states that a set H of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that is, for each x \\in X. The result can be generalized to a case when Y is locally convex and X is a [barreled space](https://bliptext.com/articles/barreled-space).\n\n### Properties of equicontinuous linear functionals\n[Alaoglu's theorem](https://bliptext.com/articles/alaoglu-s-theorem) implies that the weak-* closure of an equicontinuous subset of X^{\\prime} is weak-* compact; thus that every equicontinuous subset is weak-* relatively compact.\nIf X is any locally convex TVS, then the family of all [barrels](https://bliptext.com/articles/barrelled-space) in X and the family of all subsets of X^{\\prime} that are convex, balanced, closed, and bounded in correspond to each other by polarity (with respect to ).\nIt follows that a locally convex TVS X is barreled if and only if every bounded subset of is equicontinuous.\n\n# Equicontinuity and uniform convergence\nLet X be a compact Hausdorff space, and equip C(X) with the [uniform norm](https://bliptext.com/articles/uniform-norm), thus making C(X) a [Banach space](https://bliptext.com/articles/banach-space), hence a metric space.\nThen [Arzelà–Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem) states that a subset of C(X) is compact if and only if it is closed, uniformly bounded and equicontinuous.\nThis is analogous to the [Heine–Borel theorem](https://bliptext.com/articles/heine-borel-theorem), which states that subsets of **R**n are compact if and only if they are closed and bounded.\nAs a corollary, every uniformly bounded equicontinuous sequence in C(X) contains a subsequence that converges uniformly to a continuous function on X.\nIn view of Arzelà–Ascoli theorem, a sequence in C(X) converges uniformly if and only if it is equicontinuous and converges pointwise.\nThe hypothesis of the statement can be weakened a bit: a sequence in C(X) converges uniformly if it is equicontinuous and converges pointwise on a dense subset to some function on X (not assumed continuous).\nThis weaker version is typically used to prove Arzelà–Ascoli theorem for separable compact spaces.\nAnother consequence is that the limit of an equicontinuous pointwise convergent sequence of continuous functions on a metric space, or on a locally compact space, is continuous.\n(See below for an example.)\nIn the above, the hypothesis of compactness of X cannot be relaxed.\nTo see that, consider a compactly supported continuous function g on ****R**** with g(0) = 1, and consider the equicontinuous sequence of functions on ****R**** defined by ƒn(x) = g(x − n).\nThen, ƒn converges pointwise to 0 but does not converge uniformly to 0.\nThis criterion for uniform convergence is often useful in real and complex analysis.\nSuppose we are given a sequence of continuous functions that converges pointwise on some open subset G of **R**n. As noted above, it actually converges uniformly on a compact subset of G if it is equicontinuous on the compact set.\nIn practice, showing the equicontinuity is often not so difficult.\nFor example, if the sequence consists of differentiable functions or functions with some regularity (e.g., the functions are solutions of a differential equation), then the [mean value theorem](https://bliptext.com/articles/mean-value-theorem) or some other kinds of estimates can be used to show the sequence is equicontinuous.\nIt then follows that the limit of the sequence is continuous on every compact subset of G; thus, continuous on G.\nA similar argument can be made when the functions are holomorphic.\nOne can use, for instance, [Cauchy's estimate](https://bliptext.com/articles/cauchy-s-estimate) to show the equicontinuity (on a compact subset) and conclude that the limit is holomorphic.\nNote that the equicontinuity is essential here.\nFor example, ƒn(x) = arctan n x converges to a multiple of the discontinuous [sign function](https://bliptext.com/articles/sign-function).\n\n# Generalizations\n\n## Equicontinuity in topological spaces\nThe most general scenario in which equicontinuity can be defined is for [topological spaces](https://bliptext.com/articles/topological-space) whereas uniform equicontinuity requires the [filter](https://bliptext.com/articles/filter-set-theory) of neighbourhoods of one point to be somehow comparable with the [filter](https://bliptext.com/articles/filter-set-theory) of neighbourhood of another point.\nThe latter is most generally done via a [uniform structure](https://bliptext.com/articles/uniform-structure), giving a [uniform space](https://bliptext.com/articles/uniform-space).\nAppropriate definitions in these cases are as follows:\nWe now briefly describe the basic idea underlying uniformities.\nThe uniformity 𝒱 is a non-empty collection of subsets of\nY × Y\nwhere, among many other properties, every\nV ∈ 𝒱\n, V contains the diagonal of Y (i.e.\n{(y, y) ∈ Y}\n). Every element of 𝒱 is called an **entourage**.\nUniformities generalize the idea (taken from [metric spaces](https://bliptext.com/articles/metric-spaces)) of points that are \"r-close\" (for\nr > 0\n), meaning that their distance is 0\n, let\ndenote the set of all pairs of points that are r-close.\nNote that if we were to \"forget\" that d existed then, for any\nr > 0\n, we would still be able to determine whether or not two points of Y are r-close by using only the sets\nUr\n. In this way, the sets\nUr\nencapsulate all the information necessary to define things such as [uniform continuity](https://bliptext.com/articles/uniform-continuity) and [uniform convergence](https://bliptext.com/articles/uniform-convergence) without needing any metric.\nAxiomatizing the most basic properties of these sets leads to the definition of a [uniformity](https://bliptext.com/articles/uniform-space).\nIndeed, the sets\nUr\ngenerate the uniformity that is canonically associated with the metric space\n(Y, d)\n.\nThe benefit of this generalization is that we may now extend some important definitions that make sense for metric spaces (e.g. [completeness](https://bliptext.com/articles/complete-metric-space)) to a broader category of topological spaces.\nIn particular, to [topological groups](https://bliptext.com/articles/topological-group) and [topological vector spaces](https://bliptext.com/articles/topological-vector-space).\n\n## Stochastic equicontinuity\nStochastic equicontinuity is a version of equicontinuity used in the context of sequences of functions of [random variables](https://bliptext.com/articles/random-variables), and their [convergence](https://bliptext.com/articles/convergence-of-random-variables).","revisionCount":0},"hasMore":false,"nextCursor":null}

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