Example of fourier series diverges at least two points

Question

I know continuous function $f$ such that fourier series of $f$ diverges at one point. But, I don't know continuous function such that fourier series of $f$ diverges at least two point.

Answer 1

One of the basic properties of Fourier series is localization: If $f=0$ in a neighborhood of $t$ then the Fourier series for $f$ converges to $0$ at $t$. Hence if $f=g$ in a neighborhood of $t$ then the two Fourier series either both converge at $t$ or both diverge at $t$.

So: Given $f_1$ with a divergent series at $t_1$ and $f_2$ with a divergent series at $t_2$ concoct $g$ so that $g=f_1$ in a neighborhood of $t_1$ and $g=f_2$ in a neighborhood of $t_2$.