Let $f, g$ be continuous and not differentiable functions from $\mathbb R$ to $\mathbb R$. Let $h=f \ast g$ be the convolution of $f$ and $g$, i.e. $h(t)=\int_{-\infty}^\infty f(x).g (x-t) dx$. Is $h$ differentiable? If not, under what condition $h$ is differentiable?
My motivation example is that: $f(x)= \max ( \frac{-1}{2w} |x|+1, 0 )$ and $h=f\ast f$. From numerical computation, I observe that $h$ is differentiable. But I don't know how to prove this. It is well-known that if one of $f$ or $g$ is differentiable then $h$ is, but I couldn't find any result with respect to non-differentiability $f$ and$g$.
