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.

Proof sketch (optional): Assume not uniformly continuous; then there exists $\varepsilon_0>0$ and sequences $x_n,y_n$ with $d_X(x_n,y_n)\to 0$ but $d_Y(f(x_n),f(y_n))\ge \varepsilon_0$. By compactness, pass to a subsequence $x_{n_k}\to x$. Then $y_{n_k}\to x$ as well. Continuity forces $f(x_{n_k})\to f(x)$ and $f(y_{n_k})\to f(x)$, contradicting the fixed separation $\varepsilon_0$.