Given differentiable functions $f,g$, one can make the following statement about the derivatives of their convolution:
$(f \star g)' = f' \star g = f \star g'$
Suppose I pick $g$ as a non differentiable function such as $g(x) = |x|$, does this property still hold? (plotting $(|x| \star |x|)'$ and $(|x|' \star |x|)$ in Matlab shows different functions)
If the above property is true then by definition of convolution $f \star g' (x) = \int f(y) g'(x-y) dy$
So when can we say the convolution is not differentiable whenever $g'(x-y)$ is not differentiable?