# Convergence of Fourier series for piecewise constant function

Let $f(x)=1$ for $x\in (0,\pi)$ and $f(x)=0$ for $x\in (-\pi,0)$. Furthermore, extend $f$ to be periodic of period $2\pi$.

I calculated the Fourier series of $f$ to be $$\dfrac{1}{2}+\dfrac{2}{\pi}\sum_{n>0 \text{ odd}}\dfrac{\sin(nx)}{n}.$$

I'm wondering for which $x$ this sum converges, and to what value?

There's a theorem that if $f\in C^1$, then $s_N(x)\rightarrow f(x)$ for all $x$, where $s_N$ denotes the partial sum of the Fourier series. But here we don't even have $f$ continuous.