Let $f$ and $g$ be functions on $\mathbb{R}^n$. Let $x_0$ be a given point in the unit ball $B(0,1)$. I am looking for sufficient conditions for the convolution $ (f \ast g)(x) = \int_{B(0,1)} f(y)g(x-y) dy $ to be continuous at $x_0$.
I would appreciate simple proofs or references to proofs that conditions given in an answer are sufficient.
In my specific application, $f$ and $g$ are continuous in $B(0,1) \setminus \{0\}$ and $x_0 \neq 0$, but I would be very interested to see conditions for other (more general) situations as well.
I would also be very interested to see conditions for the situation where $B(0,1)$ is replaced by $\mathbb{R}^n$.
Thanks very much!