Let $f\in\mathit{L}(\mathbb{T})$. Suppose that
$$|f(x)-f(0)|\leq\frac{1}{(\log(|x|))^2},\;\;0\neq x\in\left[\dfrac{-1}{2},\dfrac{1}{2}\right]$$
Question
Is it true that $\lim_{n\rightarrow\infty}S_n f(0)=f(0)$?
My guess is that it is not true but i can't seem to think of a counter-example. Any hints?