Uniform continuity

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(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 f(x) and f(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, comparing the sizes of neighborhoods 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.

We have the following chain of inclusions for functions over a compact subset of the real line

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

The difference between being uniformly continuous, and being simply 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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