Continuous function on a compact set is uniformly continuous

On compact domains, continuity automatically upgrades to uniform continuity

Continuous function on a compact set is uniformly continuous

Corollary (Heine–Cantor): Let $(X,d_X)$ be a compact metric space and let $(Y,d_Y)$ be a metric space. If $f:X\to Y$ is continuous , then $f$ is uniformly continuous on $X$ .

Connection to parent theorem: This is precisely the Heine–Cantor theorem .