Finding the points of continuity of $f(x) =\sum_{n=1}^{\infty} \frac{\sin nx}{n^{1/2}}$

Question

Let $$f(x) =\sum_{n=1}^{\infty} \frac{\sin nx}{n^{1/2}}$$
How can I find the points of continuity of this function?

I know that continuity of $\frac{\sin nx}{n^{1/2}}$ doesn’t guarantee the continuity of $f(x)$.

Answer 1

Because $\sin (nx) = {\rm Im} (e^{inx})$ we have $$ f(x) = {\rm Im}\big( g(e^{ix})\big)$$ where $$ g(z) = \sum_{n=1}^\infty \frac{z^n}{\sqrt{n}}$$ This series is convergent for $z\in D =\{z\in\mathbb C: |z|\le 1, z\neq 1\}$ and as a power series, it is continuous on $D$. That means that $f(x)$ is continuous for all $x\in\mathbb R$ such that $e^{ix}\neq 1$, that is $x\notin 2\pi\mathbb Z$.