When checking whether a function is equal to its Fourier series one can use the Dirichlet conditions.
If they are satisfied, one gets that $f$ is equal to its Fourier series wherever it is continuous. At the points of dicontinuity the Fourier series converges to the mean of the left and right limit of $f$ at that point.
So far, so good!
Except:
Usually, in problems involving Fourier series, the function could be something like $x^2$ but restricted to $[0,2\pi]$ and then periodically continued on all of $\mathbb R$.
In which case, if I'm not mistaken, there are points of discontinuity at $2\pi k$ so the Fourier series would only represent the function on points not equal to $2 \pi k$. (maybe unless by coincidence, the two limits are equal so that the mean value of them is equal to each limit? but then the function would still have a spike and not be continuous there)
Am I missing something? I'm asking because I just read through an example with $f(x) = x^4 - \pi x^2$ on $[-\pi,\pi)$ and this $f$ is supposedly continuous on all of $\mathbb R$. Is this a typo?