Conditions for continuous extension of a function on an open set to its closure
Solution 1:
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.