Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

Thanks for any help .

share|cite|improve this question
Sorry for the inconvenience caused due to my editting – Ester Sep 6 '12 at 18:55
up vote 5 down vote accepted

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

share|cite|improve this answer
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

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.