Does anyone know how the graph looks like for a continuous function with Fourier series diverging at $x=0$ ?
The example due to Fejer (a variation of the du Bois-Reymond construction), is explicitly given by $$f(x)=\sum_{n=1}^\infty \frac1{n^2} \sin(2x2^{n^3})\big[\sum_{m=1}^{ 2^{n^3}} \frac{sin(mx)}m \big]$$ see eg Champeney, Handbook of Fourier Theorems, page 37. However, trying to draw $f(x)$ with basic software seems too slow, even for a few terms in the series...
Does anyone know a place where one such graph is shown?
Of course, by the Dini and Jordan convergence tests, the function will have essentially no smoothness at $x=0$, and won't be of bounded variation (so can expect lots of oscillations around $x=0$). Also, to guess its real graph one should draw the Cesaro sum of the series (or some other good approximation), as the Fourier series will show the divergent behavior stated by the theorem.