For integrable function $f$, we define $a_n=\frac{1}{2\pi} \int ^{2\pi}_0 f(x)e^{-i\pi x n} \, dx$ and the Fourier series of $f\sim\sum\limits^\infty_{-\infty}a_n e^{inx}$.
I learned from class that we need to sum this Fourier series in a symmetrical sense, i.e., $\lim_{k\to \infty}\sum^k_{-k}a_k e^{ikx}$.
There is a theorem that says if $f$ is second continuously differentiable, then $a_n=O(1/n^2)$ as $n\to \infty$ so $\sum\limits^\infty_{-\infty}|a_n|$ converges and the order of the sum in Fourier series of $f$ does not matter. But what happens if we relax the smooth condition of $f$? What happens if $f$ is discontinuous? Can someone help me with coming up with counter examples or proof?