Let $f$ be a piecewise continuous function on $[-\pi,\pi]$, with $\int_{- \pi}^{\pi} f(x) dx=0$. Define $$ F(x)=\int_{-\pi}^{x}f(y)dy.$$ Show that the full Fourier series of $F$ converge pointwise to $F$.
Definition. An infinite series $\sum_{n=1}^{\infty} f_n(x)$ converges to $f(x)$ pointwise in $(a,b)$ if it converges to $f(x)$ for each $a< x < b$. For each $a < x< b$ we have $$ \lim_{N\to \infty}\left|f(x)-\sum_{n=1}^N f_n(x)\right| \rightarrow 0 $$ Since we have $\int_{- \pi}^{x} f(x) dx=0$, for all functions $f_n$ on $[-\pi,\pi]$, they converge to $0$. Am I missing something?
Since the function $f$ is integrable, it is bounded. It follows that the function $F$ is Lipshitz, so its Fourier series converges pointwise, see, for instance, [Fich, 684]. See more on pointwise convergence of Fourier series in Wikipedia.
References
[Fich] Grigoriy Fichtenholz, Differential and Integral Calculus, v. III, 4-th edition, Moscow: Nauka, 1966, (in Russian).