Now, are there examples of functions, for which the Fourier transform converges ($\hat f $ exists for almost every $\omega \in \mathbb R$) but the integral $\mathrm{p.v.}\int_{-\infty}^\infty \hat f(\omega)e^{+i\omega t}d\omega $ does not converge (in a set with a measure bigger than zero)? or does converge, but not to $f(t)$? (in points where $f$ is continuous, for example)