an example of a continuous function whose Fourier series diverges at a dense set of points

As I mentioned in comments below, Kolmogorov's example is for a discontinuous function in $L^1$.

For a continuous function whose Fourier series diverges at all rational multiples of $2\pi$ (and hence on a dense set) see Katznelson's book: An Introduction to Harmonic Analysis Chapter 2, Remark after proof of Theorem 2.1. Note that the Fourier series of such a continuous function still converges almost everywhere by Carleson's theorem.

Actually, such an almost-everywhere divergent Fourier series was constructed by Kolmogorov.

For an explicit example, you can consider a Riesz product of the form:

$$ \prod_{k=1}^\infty \left( 1+ i \frac{\cos 10^k x}{k}\right)$$

which is divergent. For more examples, see here and here.

Edit: (response to comment). Yes, you are right, du Bois Reymond did indeed construct the examples of Fourier series diverging at a dense set of points. However the result of Kolmogorov is stronger in that it gives almost everywhere divergence.

The papers of du Bois Reymond are:

Ueber die Fourierschen Reihen

available for free download here also another one here.

Kolmogorov improved his result to a Fourier series diverging everywhere. Original papers, in French:

Kolmogorov, A. N.: Une série de Fourier-Lebesgue divergente presque partout, Fund. Math., 4, 324-328 (1923).

Kolmogorov, A. N.: Une série de Fourier-Lebesgue divergente partout, Comptes Rendus, 183, 1327-1328 (1926).