Relation between Continuity and Uniform Continuity

Question

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

Thanks for any help .

Sorry for the inconvenience caused due to my editting – Ester Sep 06 '12 at 18:55 — Ester, Sep 06 '12 at 18:55

score 5 · Accepted Answer · answered Sep 06 '12 at 12:10

5

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

answered Sep 06 '12 at 12:10

Julián Aguirre

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 06 '12 at 15:13
@HagenvonEitzen I do not know. In any case I think $X$ should be complete. – Julián Aguirre Sep 06 '12 at 18:16
@HagenvonEitzen This paper gives a positive answer to the question. – Julián Aguirre Sep 07 '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 07 '12 at 18:52
@Ricky I could get the whole paper. May be my institution has full access. – Julián Aguirre Sep 10 '12 at 08:32

score 3 · Answer 2 · answered Sep 06 '12 at 12:10

3

In all compact metric spaces continuity and uniform continuity is equivalent.

answered Sep 06 '12 at 12:10

Ralph Tandetzky

2 Answers2