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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
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up vote 5 down vote accepted

All compact metric spaces, since a continuous function on a compact metric space is uniformly continuous.

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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
    
@HagenvonEitzen This paper gives a positive answer to the question. –  Julián Aguirre Sep 7 '12 at 11:24
    
@Julian I am only getting the first page of the paper.Can you please give me the link to the full paper? –  Ester Sep 7 '12 at 18:52
    
@Ricky I could get the whole paper. May be my institution has full access. –  Julián Aguirre Sep 10 '12 at 8:32
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In all compact metric spaces continuity and uniform continuity is equivalent.

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