I am trying to show that $cl(C_c(\Omega))$ is dense in $C_0(\Omega)$ where $C_c(\Omega)$ is the set of compactly supported continuous functions $f: \Omega \rightarrow R$ and $C_0(\Omega)$ is the set of continuous functions $f: \Omega \rightarrow R$ that can be continuously extended to $\partial \Omega$ and $f |_{\partial \Omega} = 0$. As a first step I want to show that it's a subset.
Edit: $\Omega \subset R$ is a bounded open domain.
I think I have the idea but I don't know how to write it down, maybe someone can help me. Here is the idea:
Let $f \in cl(C_c(\Omega))$. Then $ \forall \varepsilon > 0 \exists g \in C_0(\Omega): || f - g ||_\infty < \varepsilon$
Assume there doesn't exist an $f^\prime$ such that $f^\prime |_{\partial \Omega} = 0$ and $f^\prime |_\Omega = f$. Say, for all such $f^\prime$, $f^\prime \geq K$ on $\partial \Omega$.
Now I want to say that at the point where $\Omega$ ends and $\partial \Omega$ starts, $g$ has to be continuous and is $0$ on $\partial \Omega$ . But $f^\prime \neq 0$ and arbitrarily close to $g$, at least on $\Omega$, therefore there would have to be a point of discontinuity on $g$ or $f^\prime$ has to be zero on $\partial \Omega$.
Assuming my reasoning is correct, how can I write this properly? Many thanks for your help.