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Relation among continuous functions
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions. Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise bounded and equicontinuous. As 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). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or locally compact space is continuous. If, in addition, fn are holomorphic, then the limit is also holomorphic. The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.

A set of functions with a common Lipschitz constant is (uniformly) equicontinuous. In particular, this is the case if the set consists of functions with derivatives bounded by the same constant. Uniform boundedness principle gives a sufficient condition for a set of continuous linear operators to be equicontinuous. A family of iterates of an analytic function is equicontinuous on the Fatou set.

The sequence of functions fn(x) = arctan(nx), is not equicontinuous because the definition is violated at x0=0.

Suppose that T is a topological space and Y is an additive topological group (i. e. a group endowed with a topology making its operations continuous). Topological vector spaces are prominent examples of topological groups and every topological group has an associated canonical uniformity. Definition: A family H of maps from T into Y is said to be equicontinuous at t ∈ T if for every neighborhood V of 0 in Y, there exists some neighborhood U of t in T such that h(U) ⊆ h(t) + V for every h ∈ H. We say that H is equicontinuous if it is equicontinuous at every point of T.

Note that if H is equicontinuous at a point then every map in H is continuous at the point. Clearly, every finite set of continuous maps from T into Y is equicontinuous.

Because every 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.

The uniform boundedness principle (also known as the Banach–Steinhaus theorem) states that a set of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that is, for each The result can be generalized to a case when is locally convex and is a barreled space.

Alaoglu's theorem implies that the weak * closure of an equicontinuous subset of is weak * compact; thus that every equicontinuous subset is weak * relatively compact. If is any locally convex TVS, then the family of all barrels in and the family of all subsets of that are convex, balanced, closed, and bounded in correspond to each other by polarity (with respect to ). It follows that a locally convex TVS is barreled if and only if every bounded subset of is equicontinuous.

Let X be a compact Hausdorff space, and equip C(X) with the uniform norm, thus making C(X) a Banach space, hence a metric space. Then Arzelà–Ascoli theorem states that a subset of C(X) is compact if and only if it is closed, uniformly bounded and equicontinuous. This is analogous to the Heine–Borel theorem, which states that subsets of Rn are compact if and only if they are closed and bounded. As a corollary, every uniformly bounded equicontinuous sequence in C(X) contains a subsequence that converges uniformly to a continuous function on X. In view of Arzelà–Ascoli theorem, a sequence in C(X) converges uniformly if and only if it is equicontinuous and converges pointwise. The 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). This weaker version is typically used to prove Arzelà–Ascoli theorem for separable compact spaces. Another 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. (See below for an example.) In the above, the hypothesis of compactness of X  cannot be relaxed. To 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). Then, ƒn converges pointwise to 0 but does not converge uniformly to 0. This criterion for uniform convergence is often useful in real and complex analysis. Suppose we are given a sequence of continuous functions that converges pointwise on some open subset G of Rn. As noted above, it actually converges uniformly on a compact subset of G if it is equicontinuous on the compact set. In practice, showing the equicontinuity is often not so difficult. For 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 or some other kinds of estimates can be used to show the sequence is equicontinuous. It then follows that the limit of the sequence is continuous on every compact subset of G; thus, continuous on G. A similar argument can be made when the functions are holomorphic. One can use, for instance, Cauchy's estimate to show the equicontinuity (on a compact subset) and conclude that the limit is holomorphic. Note that the equicontinuity is essential here. For example, ƒn(x) = arctan n x converges to a multiple of the discontinuous sign function.

Stochastic equicontinuity is a version of equicontinuity used in the context of sequences of functions of random variables, and their convergence.