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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?

Wedge
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  • You can find an answer on this post: http://math.stackexchange.com/questions/14855/an-example-of-a-continuous-function-whose-fourier-series-diverges-at-a-dense-set – tks Jan 21 '15 at 09:04

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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.