Let $\Omega \subset \mathbb{R}^n$ be an open bounded set, $H^{m}(\Omega)=W^{m,2}(\Omega)$ be the Sobolev space of all functions in $L^2(\Omega)$ with weak derivatives of order $\leq m$ also in $L^2(\Omega)$, and $C^{m}(\bar{\Omega})$ be the set of functions in $C^m(\Omega)$ such that all its derivatives of order $\leq m$ admits a continuous extension to $\bar{\Omega}$. Show that $C^{m}(\bar{\Omega}) \subset H^{m}(\Omega)$.
How should I proceed? I must prove that there is $g \in L^2(\Omega)$ such that $$\int_\Omega f \partial^\alpha_x\varphi = (-1)^{|\alpha|} \int_\Omega \partial^\alpha_xf\varphi,$$ $\forall \varphi \in C^\infty_c(\Omega)$. I know that $\bar{\Omega}$ must be compact, but that guarantees only that if $f \in C^{m}(\bar{\Omega})$, then $f$ is compactly supported in $\bar{\Omega}$, not on $\Omega$, then I can't use integration by parts. Any hints?