Consider a function $g \in L^2(-\pi,\pi)$ such that it is continuous at $x \in (-\pi,\pi)$. Prove that if the Fourier series of g converges at x then that implies g(x) is its limit.
I was thinking of applying the Fejer's Theorem theorem for this problem. My interpretation of this question is that if the Fourier series of g, converges at a particular point $x \in (-\pi,\pi)$ then the value of the convergence is g(x)?
This is the combination of two standard results. You don't even need $L^2$; being in $L^1$ is enough.
If $g\in L^1[-\pi,\pi]$ and continuous at $x$, then the Fourier series is Cesàro summable to $g(x)$. This is a form of Fejér's theorem.
If a series converges to $S$, then the Cesàro summation process also gives $S$. This has nothing to do with Fourier series; it's just that the convergence $s_n\to s$ implies $$\frac{1}{N}(s_1+\dots+s_N)\to s\quad \text{ as } \ N\to\infty$$