Show that any two different functions in $C(\Omega)$ differ as generalized functions as well.
Let $f,g \in C(\Omega)$. Since $f\ne g$, there is a $x_0 \in \Omega$ such that $f(x_0)\ne g(x_0)$. Hence there exists a neighborhood say ($U_{x_0}$). Let's say $h(x)=f(x)-g(x) \gt 0$ for all $x$ in $U_{x_0}$. Let $K$ be a compact set in $U_{x_0}$ There exists $\phi \in D(\Omega)$ such that $\phi=1$ on $K$ and $0$ outside $U_{x_0}$. Since $K$ is compact, there is a $\delta \gt 0$ such that $h(x) \gt 0$ in $k$. Then $(T_h,\phi)$ on $K$ is $\int_{k}(f(x)-g(x))\phi(x) dx \ne 0$. Hence $\int_{U_{x_0}}f(x)\phi(x)dx \ne \int_{U_{x_0}}g(x)\phi(x)dx$. Thus they differ as generalized functions as well.
The only question is how DO I get hold of the compact set $K$?? Since $\Omega \subset \mathbb{R^d}$, I can do that. What if $\Omega$ is any other space??
Thanks for the help!!