Extensions of continuous bounded functions

Question

If $u:U \rightarrow \mathbb{R}^{n}$ is bounded and continuous can $u$ always be extended such that $u \in C(\bar{U})$? and is $u$ uniformly continuous?

Answer 1

No, this is not always possible, let $n = 1$ and $U = (0,1) \subseteq \mathbb R$, define $u(x) = \sin(x^{-1})$. Then $u$ is bounded (by 1) and continuous, but is neither uniformly continuos nor extendable to $[0,1] = \bar U$.