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Uniformly Continuous


A map f from a metric space M=(M,d) to a metric space N=(N,rho) is said to be uniformly continuous if for every epsilon>0, there exists a delta>0 such that rho(f(x),f(y))<epsilon whenever x,y in M satisfy d(x,y)<delta.

Note that the delta here depends on epsilon and on f but that it is entirely independent of the points x and y. In this way, uniform continuity is stronger than continuity and so it follows immediately that every uniformly continuous function is continuous.

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 f(x)=tan(x) (for x in (-pi/2,pi/2)) and g(x)=e^x (for x in R). On the other hand, every function which is continuous on a compact domain is necessarily uniformly continuous.


See also

Continuous Function, Equicontinuous

This entry contributed by Christopher Stover

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References

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

Cite this as:

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

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