density of spaces, Lebesgue

Question

My question is the following: Is the space $C^\infty(\bar{G})$ dense in $L^\infty(G)$ ? Assume that $G \subset \mathbb{R}^n$. I know this result holds, if we have $G$ instead of $\bar{G}$. But is it true here?

Answer 1

This is not true in most reasonable cases. Here is an particular example illustrating the fenomena:

You cant approximate $\operatorname{sign}\in L_\infty([-1,1])$ function by smooth functions in $\sup$ norm because this would imply that uniform limit of countinuous functions is discontinuous, which is well known to be untrue.