TITLE: Relation between Continuity and Uniform Continuity
QUESTION [7 upvotes]: Can we define a particular metric ( except for discrete metric ) on any given set in which continuity implies uniform continuity ?
Thanks for any help .

REPLY [5 votes]: All compact metric spaces, since a continuous function on a compact metric space is uniformly continuous.

REPLY [3 votes]: In all compact metric spaces continuity and uniform continuity is equivalent.