Example of continuous function whose Fourier series doesn't converge on an uncountable dense set. According to a well-known theorem (Theorem 5.12 in Rudin's Real and Complex Analysis), there is a dense $G_\delta$ set of continuous periodic functions $f:\mathbb{R}\to\mathbb{C}$ such that the Fourier series of $f$ does not converge for all $x$ in a dense $G_\delta$ subset of $\mathbb{R}$ (hence uncountable).
This is completely mind blowing, especially the fact there is a whole dense $G_\delta$ set of such pathological functions. But is there at least one known explicit example of such a function?

An example is given in N K Bary's A Treatise on Trigonometric series Volume 1  Chapter 4 section 20.
 There are other examples of 'similar' nature in this monumental work. 
It appears that Katznelson's example is countable.
The construction is very tedious. It involves Fejer's polynomial. A detail analysis to show that the set of divergence is of second category is given in Nina Bary's book.  
