Suppose $f: \mathbb{R} \to \mathbb{R}$ is continuous, differentiable and suppose $f$ and $f'$ are both summable. Then
$$\mathcal{F}(f')(x) = -ix \mathcal{F}(f)(x)$$
(Here $\mathcal{F}$ represents fourier transform, i.e. $\mathcal{F}(f)(x) = \int f(t) e^{itx} dt$)
My question is: what goes wrong if $f$ is only piece-wise continuously differentiable? That is, I am looking for some counter example of how the above formula fails in this case.