how a discontinuous function converges to Hermite- Fourier series?

Question

I have the proof using a text that if a function $f (x)$ is square integrable with weight function $e^{-x^2}$ and also is piecewise continuous, then $f (x)$ converges to $$f(x)=\underset{n=0}{\overset{\infty}\sum}c_nH_n(x),~~~~~-\infty<x<\infty$$ where $~~~~~$ $c_n=\frac{1}{2^nn!\sqrt{\pi}}\int_{-\infty}^\infty e^{-x^2}f(x)H_n(x)dx,~~~n=0,1,2,...\\$ and $H_n(x)$ are the hermite polynomials. It is clear, but I do not understand what happens when the function is discontinuous in $x$, according to the book $f (x)$ converges to $\frac{f(x+0)+f(x-0)}{2}$. How could you prove this?.