Exercise 4.4.14

Let $f$ be continuous on $K$, and choose $𝜖>0$. We can create the open cover $\{V_{{\delta }_x∕2}(x):x\in K\}$ where ${\delta }_x$ is chosen so every $y\in V_{\delta }(x)$ has $|f(x)-f(y)|<𝜖∕2$. Now since $K$ is compact there exists a finite subcover $O=\{V_{{\delta }_1∕2}(x_1),\dots ,V_{{\delta }_n∕2}(x_n)\}$ with $K\subseteq {\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{k=1}^n{V}_{{\delta }_k∕2}(x_k)$.

Let $\delta =\min <!--\; FUNCTION\; APPLICATION\; -->\{{\delta }_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,{\delta }_n\}∕2$ and choose arbitrary $x\in K$. Since $O$ is an open cover of $K$, there must be some $V_{{\delta }_i∕2}(x_i)\ni x$. Now suppose $y\in K$ so that $|x-y|<\delta$. Then

Also, $|x-x_i|<{\delta }_i∕2<{\delta }_i$. Since $f$ is continuous at $x_i$, this implies $|f(y)-f(x_i)|<𝜖∕2$ and $|f(x)-f(x_i)|<𝜖∕2$, so by the Triangle Inequality $|f(x)-f(y)|<𝜖$.