Continuous periodic functions not approximable by a Fourier series

Question

What is an example of a continuous periodic function that is not the limit of any Fourier series? If not, is there an more or less elementary proof?

Answer 1

In part, there is some potential imprecision of language that confuses beginners here. I know it confused me. Plus, a surprising fact:

First, it has been known since at least since Fejer that every continuous function is a uniform-pointwise limit of finite Fourier series $\sum_{|m|\le N} c_{N,n} e^{2\pi inx}$.

More specifically, Fejer gave a formula for $c_{N,n}$ in terms of the Fourier coefficients $\widehat{f}(n)$ of the function $f$ itself...

The seeming paradox is that $c_{N,n}$ definitely cannot be $\widehat{f}(n)$ in general! That is, "the Fourier series" of $f$'s finite subsums are in general not the sequence of Fourier sums that converge uniformly to $f$. Crazy, seems to me.