ANALYSIS 2026-05-12

Heine-Cantor Theorem

Theorem

(Heine–Cantor Theorem)

If $K$ is a compact metric space and $f:K\to M$ is continuous, then $f$ is uniformly continuous.

Ordinary continuity allows $\delta$ to depend on the point $x$ . Uniform continuity demands a single $\delta$ that works everywhere on $K$ . Compactness is exactly the hypothesis needed to promote the former to the latter.