Given two sets of finite measure in $\mathbb{R}$ say, $E$ and $F$, and their characteristic functions $\chi_E$ and $\chi_F$, can somebody show that $\chi_E\ast\chi_F(x)$ (the convolution) is a continuous function of $x$? This is a qual problem from an old qual that I'm studying, and I cannot figure it out. If we were dealing with continuous functions or mollifiers or something it would be straightforward, but what if the sets $E$ and $F$ are somehow pathological, like the Cantor set, or something like that?
Thanks!