Given any countable subset $A$ of $[-\pi, \pi]$, there exists a function $f \in C[-\pi, \pi]$ whose Fourier series diverges for all $x$ $\in$ $A$.

Question

Given any countable subset $A$ of $[-\pi, \pi]$, there exists a function $f \in C[-\pi, \pi]$ whose Fourier series diverges for all $x \in A$.

By Du Bois Reymond theorem we get a continuous function whose Fourier series diverges on a countable dense subset of $[-\pi, \pi]$. How to get divergence for a general countable set?