Determine convergence or divergence of Fourier series of a continuous function on $\left[-\pi,\pi\right]$ at $x=0$

Question

Function $f$ is defined as following:
$$f\left(x\right)=\frac{\cos\frac{1}{x}}{\log\frac{\left|x\right|}{4}}\quad x\ne 0,\\f\left(0\right)=0$$
$f$ is a continuous function on $\left[-\pi,\pi\right]$.
However, since $f$ does not satisfy Dini condition at $x=0$ and is not a BV function, it seems hard to tell whether its Fourier series converges to $f\left(0\right)$ at $0$.
Someone has any ideas?