How would you access the following problem:
(a) Show that for every $s \in \mathbb{Q}$ the function $$f: \mathbb{C}^* \rightarrow \mathbb{C}$$ $$ f(z) := \frac{\overline z}{\vert z \vert ^s}$$ is continuous
(b) for which $s \in \mathbb{Q}$ can we continuously continue $f$ in the origin? Is this continuation unique?
I'm pretty lost with this problem so im thankful for any kind of help and instruction thanks!
(a) complex conjugation, absolute value, rational power, division are all continuous, hence so is $f$.
(b) Uniqueness is clear because $0$ is in the closure of $\mathbb C^\times$. For existence, consider cases $s>1$, $s=1$, $s<1$ separately.