Show that a convolution of two functions solves an ODE

Question

Given a function $f\in C_c(\mathbb{R})$, meaning $f$ is continuous with compact support in $\mathbb{R}$, and function $\Phi(x) = \frac12|x|$, show that the convolution $u=f \ast \Phi$ is well defined and a solution to $$u^{\prime\prime}=f(x) \ \ \forall x\in\mathbb{R}.$$ I really don´t know how to tackle this problem. In our lecture we went over a proposition that defines the derivative of a convolution, but since $\Phi$ is not continuously differentiable over $\mathbb{R}$ and f is not explicitly given I don´t know how to use it here. Help would be very much appreciated, thanks in advance!