Consider a function $f_p : \mathbb{R} \to \mathbb{R}$ which is continuously differentiable in $(0,2\pi)$ except at two points $x = x_c$ and $x = x_o$. At $x = x_c$, $f_p(x)$ has a jump discontinuity. At the point $x = x_o$, $f_p'(x_o^+)$ and $f_p'(x_o^-)$ exist and are not equal.
Now define a function $f$ equal to $f_p$ on $(0,2\pi)$ and let it be a periodic function with period $2\pi$. Let $\hat{f}_k$ be the Fourier series coefficients of $f$. What I would like to know is whether the Fourier series defined by the coefficients $ik\hat{f}_k$ converge to $\frac{f'(x_o^+)+f'(x_o^-)}{2}$ at $x = x_o$ ?