I am having trouble with the following problem:
Let $\Omega \subseteq \mathbb{R^n}$ be open, bounded with piecewise smooth boundary. Let $\Omega_j, j = 1...J$ be open and disjoint subsets of $\Omega $ with piecewise smooth boundaries, such that $ \bar\Omega= \bar\Omega_1 \cup ...\cup\bar\Omega_J$ Prove that any $u|_{\bar\Omega_j} \in C^1(\bar \Omega_j)$ and $u \in H^1(\Omega)$ belongs to $C(\bar\Omega)$. You can use that the space $C^\infty_0(\Omega)$ is dense in $L^2(\Omega)$.
I know that the suggested density is equivalent to the existence of a sequence of functions in $C^\infty_0(\Omega)$ that converges in $L^2(\Omega)$ norm to an element of $L^2(\Omega)$, for any function of $L^2(\Omega)$. But I don't see how that helps in the problem.
On the other hand if $u \in H^1(\Omega)$, then $u \in L^2(\Omega)$ and $u_{x_i}\in L^2(\Omega)$
Also $u|_{\bar\Omega_j} \in C^1(\bar \Omega_j) \subseteq C^0(\bar \Omega)$
Any help will be appreciated