Let $\alpha >1$ and function $f$ defined as
\begin{equation*} f(x) = \left\{ \begin{array}{ll} (\text{log}\frac{1}{|\text{sin}(x)|})^{-\alpha} & \textrm{$x\neq k\pi,k\in\mathbb{Z}$}\\ 0 & \textrm{$x=k\pi,k\in\mathbb{Z}$}\\ \end{array} \right. \end{equation*}
Prove (or disprove) that the Fourier series of $f$: \begin{equation*} s_N(f;x)=\sum_{n=-N}^N \hat{f_n}e^{inx} \end{equation*} converges to $f$ pointwisely.
Note that the function $f$ goes to $\infty$ when $x=\frac{k\pi}{2}$ for any $k\in\mathbb{Z}$ and $f$ is differentiable except those points. Can one just apply the theorem that $f\in C^1$ then $S_N(f;x)\rightarrow f(x)$ for all $x\in\mathbb{R}$? Also it is noticeable that $f$ is not in $L^1(\mathbb{R})$.