# Differentiability of convolution

• I
If f and g are continuous functions on the right half-line, [0,∞], then f✶g, the convolution of f and g, is defined by
f✶g(x) = ∫[0,x] f(t)g(x-t)dt.
I would like to know if f✶g is a differentiable function of x.
If, for example, g(t) = 1 for t ≥ 0 then f✶g(x) = ∫[0,x]f(t)dt has a derivative equal to f(x). But what about in general?

micromass
Staff Emeritus
Homework Helper
If either ##f## or ##g## is differentiable, then so is the convolution.

If either ##f## or ##g## is differentiable, then so is the convolution.
Great, thanks. I now see that. But what happens if both f and g are nowhere differentiable?

Samy_A