I want to prove that for any $g \in C_0^\infty$, $g\colon[0,\infty) \times \mathbb R \rightarrow \mathbb R$, and $g = g(t,x)$,
$\sup_{x\in \mathbb R} |g(t,x)| \leqslant \int_\mathbb R \hspace{2mm} \left(\left|g(t,x)\right| + \left| \frac{\partial g(t,x)}{\partial x} \right| \right) \hspace{2mm}dx$
for any $t$.