Heine-Cantor Theorem

Definition

Definition in Real Numbers

A Continuous Function $f$ on a closed and bounded interval $[a,b]$ is uniformly continuous on the domain $[a,b]$.

Definition in Metric Space

Consider a Continuous Function $f:(X,{\mathcal{T}}_X,d_X)\to (Y,{\mathcal{T}}_Y,d_Y)$ between two metric spaces. If $X$ is compact, then $f$ is uniformly continuous.