Let $f$ be a function that is $2\pi$-periodic satisfying $f(x) = x^2$ for $-\pi < x < \pi$ and $f(\pi) = 100$ i.e. it's repeating $x^2$ and $100$ on both endpoints. This function is continuous on $(-\pi, \pi)$, but can we compute its fourier coefficients?
i.e. is it integrable on $[-\pi, \pi]$?