I have the following lemma without proof:
Lemma. Let $f, g \in L^\infty(\mathbb R^n)$ with compact supports. Then $f \ast g \in C(\mathbb R^n)$.
Is this even true?
I get this: $$ \begin{align*} \vert (f \ast g)(x) - (f \ast g)(x') \vert &\leq \int_{\mathbb R^n} \vert f(x - y) - f(x' - y) \vert \vert g(y) \vert \, \mathrm dy \\ &\leq \Vert g \Vert_{L^\infty} \int_{\mathbb R^n} \vert f(x - y) - f(x' - y) \vert \, \mathrm dy \end{align*} $$
Is there a way to estimate this further?