Let $f,g : \mathbb R \rightarrow \mathbb R$ continuous and compactly supported. I want to show that $f*g$ is continuous and compactly supported. I am 100% sure how to do it.
I began as follows:
\begin{align*} |(f*g)(x)-(f*g)(x')| \leq \int_{\mathbb R} |f(y)||g(x-y)-g(x'-y)| dy \\ \leq M \int_{\mathbb R} |g(x-y)-g(x'-y)| dy \end{align*} where $f$ is bounded by $M$ on $\mathbb R$. What must I do now ?