Heine–Cantor Theorem

Heine–Cantor Theorem: 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 $X$; i.e., $\forall \varepsilon>0\;\exists \delta>0\;\forall x,y\in X:\ d_X(x,y)<\delta \Rightarrow d_Y(f(x),f(y))<\varepsilon.$

This upgrades pointwise continuity to uniform control, and is essential for exchanging limits and integrals on compact domains and for approximation arguments.