Uniform continuity

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In mathematics, a function ƒ is uniformly continuous if, roughly speaking, it is possible to guarantee that ƒ(x) and ƒ(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between ƒ(x) and ƒ(y) cannot depend on x and y themselves. For instance, any isometry (distance-preserving map) between metric spaces is uniformly continuous.

Every uniformly continuous function between metric spaces is continuous. Uniform continuity, unlike continuity, relies on the ability to compare the sizes of neighbourhoods of distinct points of a given space. In an arbitrary topological space this may not be possible. Instead, uniform continuity can be defined on a metric space where such comparisons are possible, or more generally on a uniform space.

The equicontinuity of a set of functions is a generalization of the concept of uniform continuity.

Contents

Definition for functions on metric spaces

Given metric spaces (Xd1) and (Yd2), a function ƒ : X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every xy ∈ X with d1(xy) < δ, we have that d2(ƒ(x), ƒ(y)) < ε.

If X and Y are subsets of the real numbers, d1 and d2 can be the standard Euclidean norm, | · |, yielding the definition: for all ε > 0 there exists a δ > 0 such that for all xy ∈ X, |x − y| < δ implies |ƒ(x) − ƒ(y)| < ε.

The difference between being uniformly continuous, and simply being continuous at every point, is that in uniform continuity the value of δ depends only on ε and not on the point in the domain.

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