I've read this in my notes
Let $f:\mathbb R\to \mathbb R$ be $2\pi$-periodic, and piecewise continuous with jump discontinuities such that $\displaystyle f(a)=\frac{1}{2}\frac{f(a^+)-f(a^-)}{2}$ .
Then the series $\displaystyle\sum_{|n|\geq0}c_ne^{int}$ converges to $f$ with respect to $L^2$ convergence.
Nevertheless, there need not exist some $x$ such that $\displaystyle\lim_{n\to \infty} \sum_{|n|\geq0}c_ne^{inx}=f(x)$
The last sentence baffles me, how is it possible that the Fourier series converges pointwise nowhere ? Can someone give an example ?
I know that stronger hypotheses counter this statement (if one assumes that $f$ is piecewise smooth for example).
Your notes are messed up; maybe you dozed off while the lecturer changed the assumptions on $f$? :) As others said, $f\in L^2$ implies that the Fourier series converges to $f$ a.e., by Carleson's theorem.
The way your statement is set up is also weird. Setting $$\displaystyle f(a)=\frac{1}{2}\frac{f(a^+)-f(a^-)}{2}\tag1$$ at the jump points of a piecewise continuous function has no effect on $L^2$ convergence of Fourier series. Generally, redefining a function on any set of measure zero does nothing in the context of $L^p$ spaces.
The reason one often sees (1) in this subject is an old theorem of Dirichlet: if $f$ has bounded variation and satisfies (1) at every point, then the Fourier series converges to $f$ pointwise. (Strictly speaking, Dirichlet proved this for functions with a finite number of extrema, since BV was not invented yet.)