I know that the series $\sum_{n=1}^\infty \sin(nx) / n^\alpha$, with $0 < \alpha < 1$, converges for $x \in [0,2\pi]$.

I'm trying to understand if the function $f(x) = \sum_{n=1}^\infty \sin(nx) / n^\alpha$ is continuous on $[0,2\pi]$. I think not. In particular, I think that $\lim_{x \to 0^+} f(x) = +\infty$ while, obviously, $f(0) = 0$. But unfortunately, I'm not even able to find a sequence $x_n \to 0^+$ such that $f(x_n) \not\to 0$.

Does anyone have any suggestions? Thank you very much.