Suppose $U$ is a open set in $\Bbb R^n$, and suppose $f\colon U\to \Bbb R$ is a continuous function. Suppose that $f$ is uniformly continuous on every bounded subset of $U$.
Question: Can $f$ be continuously extended to the closure of $U$ in $\Bbb R^n$?
Yes: fix a point $x_0$ in the boundary of $U$. Then $f$ is uniformly continuous on $U\cap B(x_0,1)$. If we take a sequence $\{x_k\}$ converging to $x_0$, then the sequence $\{f(x_k)\}$ is Cauchy. We have to check that the limit doesn't depend on the chosen sequence.