[Math] An example of a function which is not piecewise continuous, but has Fourier series

Question

Would you Please give an example of a function which is not piecewise continuous, but has Fourier series?

It means that the coefficient in the Euler-Fourier formulas can be computed. In fact, the definite integrals exist.

Answer 1

Consider the characteristic function $c$ of the rationals; it's 1 at every rational, 0 at every irrational. Its FT is the constant function 0. Because $c$ is equal, almost everywhere, to the constant function $0$, they have the same FT.