Uniform continuity

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

\epsilon > 0

there exists

\delta >0

such that for all

x,y\in M

with

d(x,y)<\delta

, we have that

d(f(x),f(y))<\epsilon

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 (x_n) is a Cauchy sequence and f is a uniformly continuous function, then (f(x_n)) 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.

Mathematical analysis

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Definition

Properties

Generalization to topological vector spaces

Generalization to uniform spaces