Uniform continuity

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In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x affect small changes in the output f(x) ("continuity"), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but not on x itself ("uniformity").

Continuity itself is a local property of a function—that is, a function f is continuous, or not, at a particular point, and when we speak of a function being continuous on an interval, we mean only that it is continuous at each point of the interval. In contrast, uniform continuity is a global property of a function, and it is fundamentally incorrect to declare a function to be uniformly continuous at a point. Uniform continuity can be defined on an interval, but not at a point. In fact, any function continuous on a closed bounded interval is also uniformly continuous on that interval.

Uniform continuity, unlike continuity, is meaningless in an arbitrary topological space, since it relies on comparing sizes of open sets at distant points of a space. Instead, uniform continuity can be defined in a metric space where such comparisons are possible, or in a uniform space.

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[edit] Definition

Given metric spaces (X,d1) and (Y,d2), if M \subseteq X and N \subseteq Y then a function f : M\to N is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x,y \in M 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 Euclidian norm, |\cdot|, yielding the definition: for all ε > 0 there exists a δ > 0 such that for all x,y \in X, | x − y | < δ implies | f(x) − f(y) | < ε.

[edit] Properties

Every uniformly continuous function is continuous, but the converse is not true. Consider for instance the function f(x) = 1/x with domain the positive real numbers. This function is continuous, but not uniformly continuous, since as x approaches 0, the changes in f(x) grow beyond any bound.

If M is a compact metric space and N a metric space, then every continuous f : M → N is uniformly continuous (this is the Heine-Cantor theorem). In particular, if a function is continuous at every point of a closed bounded interval, it is uniformly continuous on that interval.

Every Lipschitz continuous map between two metric spaces is uniformly continuous.

Every uniformly continuous function is Cauchy-continuous, i. e. if (xn) is a Cauchy sequence contained in the domain of f (though perhaps not convergent in the domain of f) and f is a uniformly continuous function, then (f(xn)) is also a Cauchy sequence.

If a function f is uniformly continuous over a finite interval (a,b), then f is also bounded over (a,b).

[edit] Generalization to topological vector spaces

In the special case of two topological vector spaces V and W, the notion of uniform continuity of a map f:V\to W becomes : for any neighborhood B of zero in W, there exists a neighborhood A of zero in V such that v_1-v_2\in A implies f(v_1)-f(v_2)\in B.

[edit] Generalization to uniform spaces

The most natural and general setting for the study of uniform continuity are the uniform spaces. A function f : X → Y between uniform space is called uniformly continuous if for every entourage V in Y there exists an entourage U in X such that for every (x1, x2) in U we have (f(x1), f(x2)) in V.

In this setting, it is also true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences and that continuous maps on compact uniform spaces are automatically uniformly continuous.