Uniform continuity

(Redirected from Uniformly continuous)

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"). In other words, the function never has an infinite slope.

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. A function is uniformly continuous, or not, on an entire interval, and may be continuous at each point of an interval without being uniformly continuous on the entire interval.

Definition

Given a metric spaces X with distance function $d(\cdot)$, if $M,N\in X$ 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 all $x,y\in M$ with d(x,y) < δ, we have that d(f(x),f(y)) < ε.

If X is the domain of real numbers, $d(\cdot)$ can be substituted for the standard Euclidian norm, $|\cdot|$, yeilding the definition: if for all ε > 0 there exists a δ > 0 such that | xy | < δ implies | f(x) − f(y) | < ε.

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, 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.

If (xn) is a Cauchy sequence and f is a uniformly continuous function, then (f(xn)) is also a Cauchy sequence.

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 W such that $v_1-v_2\in A$ implies $f(v_1)-f(v_2)\in B$.

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.

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