Uniform Continuity of Continuous Functions on Compact Metric Spaces

Abstract

(This is intended as a classroom note) We give a direct proof of the fact that a continuous function on a compact metric space is automatically uniformly continuous. The proof is based on standard theorems from metric spaces: The inverse image of a closed set under a continuous function is continuous, the product of two compact metric spaces is compact, and every real valued continuous function attains a minimum on a compact set.