For $f \in \mathcal{D}'(\mathbf{R})$(The dual space of test function on $\mathbb{R}),$ I want to prove that there exists a solution $u \in \mathcal{D}'(\mathbf{R})$ such that $u$ is the solution of the equation $u'=f.$
My attempt is to use the existence of fundamental solution, i.e there exists $E \in \mathcal{D}'(\mathbf{R})$ such that $E'=\delta,$ then $E*f$ is the solution. But the problem is, if $f$ doesn't have compact support, then convolution $E * f$ makes no sense.
Can anybody help me?