# Relation between Continuity and Uniform Continuity

Can we define a particular metric ( except for discrete metric ) on any given set in which continuity implies uniform continuity ?

Thanks for any help .

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Sorry for the inconvenience caused due to my editting –  Ester Sep 6 '12 at 18:55

It feels as if the converse holds: If $X$ is a metric space with no isolated points and every continuous $f:X\to\mathbb RR$ is uniformly continuous, then $X$ is compact. Is that true? –  Hagen von Eitzen Sep 6 '12 at 15:13
@HagenvonEitzen I do not know. In any case I think $X$ should be complete. –  Julián Aguirre Sep 6 '12 at 18:16