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The partial sum of the Fourier series for the function $f(t)=(t-\pi)\chi_{\left(0,2\pi\right)}$ is $$ S_n(t)=-2 \sum_{k=1}^{n} \frac{\sin kt}{k} $$

We saw a theorem which states that the Fourier series of a function of bounded variation converges to $\frac12\left(f(t^+)+f(t^-)\right)$ when $t$ is a point of discontinuity of $f$.
But as $S_n(0)=0$, $S_n(0) \to f(0)=0$, not $-\pi/2$.

Is it a contradiction to the theorem?

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up vote 4 down vote accepted

Note that the value $f(0)$ is not relevant since $0$ is a discontinuity point of $f$. Once the function $f$ is extended to a function $g$ defined on the whole real line with period $2\pi$, as it should be, then $g(0^+)=f(0^+)=-\pi$ and $g(0^-)=f((2\pi)^-)=\pi$. Hence $\frac12(g(0^+)+g(0^-))=0=\lim\limits_{n\to\infty}S_n(0)$, as the theorem says.

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