Many of the Fourier series problems I deal with right now are with discontinuous functions. Many times the integrals involved have to be separated because there are discontinuities.
However this is making my head hurt, why do the theorems work if the sums of trigonometric functions are continuous and the original function isn't? Namely, the theorems depend on the equality, and there isn't really any equality here.
Perhaps I'm incorrectly assuming that a Fourier series expansion is continuous, but anyway, the problem of the equality I mention still stands.
Edit: I mean specifically jump discontinuities. I think this is important for my point. Because intuitively I think that the Fourier series should have the same type of discontinuity as the original function.