Example of continuous periodic function with divergent Fourier series Let $f:[-\pi,\pi]\to\Bbb C$ be continuous such that $f(-\pi)=f(\pi)$. Let $a_n = \displaystyle \int_{-\pi}^\pi f(\theta) \overline{\exp(in\theta)} \frac{\mathrm d\theta}{2\pi}$ for $n \in \Bbb Z$ be its Fourier coefficients. It is clear that the Fourier series $\displaystyle \sum_{n \in \Bbb Z} a_n \exp(in\theta)$ converges to $f(\theta)$ almost everywhere, and that for every null set $E$ there is such an $f$ such that the Fourier series diverges exactly on $E$.
So, in particular, there is such an $f$ such that its Fourier series diverges at $\theta=0$.
However, I searched examples of such functions to no avail, since they are behind paywalls. The closest I got to such an example is in the article Beispiele Stetiger Funktionen mit Divergenter Fourierreihe ("Examples of Continuous Functions with Divergent Fourier Series"), where it is said:

Bekanntlich hat P. du Bois-Reymond zuerst die Existenz einer überall 
  stetigen Funktion erwiesen, deren Fouriersche Reihe an einer Stelle divergiert.
Herr H. A. Schwarz gab dann ein einfacheres Beispiel.

translated:

"It is well known that P[aul] du Bois-Reymond was the first to demonstrate the existence of an everywhere continuous function with a Fourier series that diverges at a point.
Then, H[ermann] A[mandus] Schwarz gave an easier example."

So my question is, could one give an explicit example of such an $f$ whose Fourier series diverges (only) at $\theta=0$?
 A: One such example (Fejér) from $[0,\pi]$ to $\Bbb R$ is given[1]: $$f(x)=\sum\limits_{k=1}^\infty\frac{\sin\left[(2^{k^3}+1)\frac x2\right]}{k^2}$$ which is even and periodic with period $2\pi$. Since $$\left\vert\frac{\sin\left[(2^{k^3}+1)\frac x2\right]}{k^2}\right\vert\le\frac1{k^2}$$ by the Weierstrass $M$-test, $f$ is convergent and continuous (sum is bounded above by $\pi^2/6$). In the proof, let $$S_n=\frac\pi2\sum_{k=0}^n a_k$$ where $a_k$ are the coefficients of the Fourier series. Also let $$\lambda_{n,m}=\int_0^{\pi} \sin \left[ (2m + 1) \frac{t}{2} \right] \cos nt \ dt \text{ and } \sigma_{n,m} = \sum_{k=0}^n \lambda_{k,m},$$ and it is proven that $$S_{2^{p^3-1}} \ge \frac{1}{p^2} \sigma_{2^{p^3-1},2^{p^3-1}} \ge \frac{1}{2p^2} \ln(2^{p^3-1}) = \frac{p^3-1}{2p^2} \ln 2$$ which clearly tends to $+\infty$ as $p\to\infty$. This means that the sequence $(S_n)$ diverges and the Fourier series of $f(x)$ at zero is also divergent.

 Reference 
 [1] Merx. J-P. (2016). A Continuous Function With Divergent Fourier Series. Available from: http://www.mathcounterexamples.net/continuous-function-with-divergent-fourier-series/. 
