Let $f$ be a real valued $2\pi$ periodic function on $\mathbb R$, integrable on compact intervals. Let $f$ be differentiable at $x_0$. We define $$ g(x)=\frac{f(x_0+x)-f(x_0-x)}{2 \sin x} $$ for $x\in (0,\pi)$.
This $g$ has the limit $g(0^+)=f'(x_0)$ because $$ g(x)=\frac{1}{2} \left( \frac{f(x_0+x)-f(x)}{x}+\frac{f(x_0-x)-f(x)}{-x}\right)\frac{x}{\sin x} $$ I don't know about the limit $g(\pi^-)$. Is $g$ integrable on $(0,\pi)$?