Let $f : \overline{\Omega} \subset \mathbb{R}^{N} \to \mathbb{R}$ be a $C^{2}(\overline{\Omega})$ function. Can we always construct a sequence $f_{n}$ such that $f_{n} \to f$ uniformly in $\overline{\Omega}$? In this case, $\Omega$ is a bounded open set. If I weaken the condition to $C(\overline{\Omega})$, can I also construct such a sequence as well?
Any help will be much appreciated!