Suppose $f$ equals $-1$ on $[-\pi, 0]$ and $+1$ on $(0, \pi)$. Prove that $S_{N} f(x) \rightarrow f(x)$ if $x \in(-\pi, 0) \cup(0, \pi)$ and that $$ \lim _{N \rightarrow \infty} S_{N} f(0)=0 $$ where we define $S_{N}$ : $$ \begin{array}{c} S_{N} f(x)=\frac{1}{\sqrt{2 \pi}} \sum_{|n| \leq N} c_{n} e^{i n x} \\ c_{n}=\frac{1}{\sqrt{2 \pi}} \int f(y) e^{-i n y} d y \end{array} $$ I think I was able to show the second part and I find $S_{N} f(0)=0$ for every $N$!
But for the first part I have some problems to show the pointwise convergence. I know the theorem in the link pointwise convergence But I dont know that we can use it for this problem or not.
Hint: If $f$ is differentiable at $x,$ then $S_N(f,x)\to f(x).$ You can derive this by using the Dirichlet kernel representation of $S_N(f,x).$