A Fourier series converges to the function a.e. so the zero Fourier series converges to this a.e. as is a null set. That is this function does have a Fourier series and it converges to the function a.e. as required.
(even the Fourier series of a function with a step discontinuity does not converge to the function at the step without a bit of jiggery pokery about what the value of the function is at the step)
RonL
Thanks. I know the function is called the Dirichlet function and I guess this bounded function cannot be expressed into a Fourier series, but not quite sure since no books I read tell the fact.
I wonder if examples like this motivated Dirichlet to propose the Dirichlet Conditon for Fourier series?