Convolution of continuous function with $\mathcal{C}^{1} $ function

Question

I'm having difficulty with the following problem:

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$ such that $\mbox{Supp}\left(f\right)$ is compact and let $g:\mathbb{R}\to\mathbb{R}$ be continuously differentiable. I need to show that the convolution $f*g$ is also continuously differentiable and that the derivative of $f*g$ is $f*g^{'}$.

I am quite stumped.

Answer 1

Write the differential quotient for $f * g$: you get two integrals. Change variables in an integral so that you can put the integrals together and collect $f$ inside the integral. See the incremental quotient of $g$ appear inside the integral. Pass to the limit.

Answer 2

For the first part, both your functions are summabel $L_1(\Bbb R)$.

Now have a look at http://people.math.gatech.edu/~heil/6338/summer08/section4c_convolve.pdf