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

) and
. 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


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


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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