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In mathematical analysis, a function f(x) is called uniformly
continuous if, roughly speaking, small changes in the input x effect 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").
Definition
The formal definition is as follows: a function f : M -> N between metric spaces is called uniformly continuous if for every real number ε > 0 there exists a number δ > 0 such that for all
x1, x2 in M with d(x1, x2) < δ,
we have d(f(x1), f(x2)) < 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.
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 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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