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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.

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You have a theorem for this type of case. When a function is piecewise differentiable, having only finitely many jump discontinuities, the Fourier series converges at points of differentiability and converges to the average of the left and right limits at the jump discontinuity.

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    $\begingroup$ Actually, as long as the function is square integrable, it converges almost everywhere to the function $\endgroup$ – user113529 Dec 12 '13 at 3:29

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