To ensure condition 1., you really need the function to be continuous, so that Fejér's theorem applies. But then the Fourier series has a strong tendency to converge pointwise. However, it is possible for the Fourier series of such a function to diverge at some points. There is an outline of a non-constructive proof of this here, but it's probably not easy to give concrete examples. This is a difficult area, dominated by Carleson's famous proof in 1966 that the Fourier series of a continuous function converges almost everywhere.