A map

*
from a metric space
*
to a metric space
*
is said to be uniformly continuous if for every
*
0" />, there exists a
*
0" /> such that
*

Examples of uniformly continuous functions include Lipschitz functions and those satisfying the Hölder condition. Note however that not all continuous functions are uniformly continuous with two very basic counterexamples being

*
(for
*
) and
*
(for
*
. On the other hand, every function which is continuous on a compact domain is necessarily uniformly continuous.

You are watching: Uniformly continuous vs continuous


SEE ALSO: Continuous Function,Equicontinuous

This entry contributed by ChristopherStover


REFERENCES:

Carothers, N.L. RealAnalysis. New York: Cambridge University Press, 2000.


CITE THIS AS:

Stover, Christopher. "Uniformly Continuous." From couchsurfingcook.com--A couchsurfingcook.com Web Resource, created by Eric W. Weisstein. https://couchsurfingcook.com/UniformlyContinuous.html


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