Let $G\subseteq \mathbb{R}^N$ be an open set. I want to show that $C_0^\infty (G) \subseteq C(\bar{G})$, where $C_0^\infty (G)$ represents the smooth functions with compact support on $G$, and $C(\bar{G})$ represents the continuous functions on $\bar{G}$.
My idea is to consider a function $f \in C_0^\infty (G)$, and using the compact support of $f$, show that for all $A \subseteq \bar{G}$, $f(\bar{A}) \subseteq \bar{f(A)}$.