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?