As far as I know, there are theorems like Jordan-Dini’s that ensure pointwise convergence of the Fourier Series of a periodic function to the function itself given that one-sided limits and derivatives exist and are finite at a point.
But what if the function is something like the periodic extension of $f(x)=\sqrt{x+\pi}$, which has vertical tangency? Or $f(x)=(x+\pi)\sin(\frac{1}{x+\pi})$, which isn`t differentiable at $x=-\pi$? Both are in $L^2([-\pi,\pi])$ as they are continuous, but how can you deal with them?