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?