Is it true that if $f\in C^1_c(\mathbb{R}^n)$ then $f\in W^{1,p}(\mathbb{R}^n)$, $1\leq p <\infty$?
Well, I know that $W^{1,p}(\mathbb{R}^n)=\overline{C^\infty(\mathbb{R}^n)}^{{\lVert\cdot\rVert}_{W^{1,p}}}$, $1\leq p< \infty$.
I was trying to use the following properties of mollification:
If $f\in C(U)$, then $$ f^\varepsilon=\eta_\varepsilon* f\rightrightarrows f $$ uniformly on compact subsets of $U$,
and $f^\varepsilon\in C^\infty(U_\varepsilon)$ for every $\varepsilon>0$, where $U_\varepsilon=\{x\in U:\text{dist}(x,\partial U)>\varepsilon\}$.
Yes, it is true and it follows directly from the fact that any $C^1$ function is also weakly differentiable and its weak derivatives are its classical derivatives (which follows straight from the definition of the weak derivative). Since in your case $f$ and its derivatives have compact support, they are also in $L^p$.