Is $f(z)=z\sin(1/z)$ continuous for $z\rightarrow 0$?

Question

I have $$f(z)=z*\sin(1/z)$$ for $z\neq 0$ and $$f(0)=0$$ to see whether the function is continuous or discontinuous at 0, $z\rightarrow 0$ where $z\in \mathbb{C}$. My first idea was to express the function as series but didn´t get far. Then I tried to transform the function so I could get a nice form but since the sine function is not bounded for the complex numbers I couldn´t get the limit. Is there any way to show it or prove it?

Answer 1

Take $x > 0$ : $$f(-ix) = -ix\sin(\frac{1}{-ix}) = -ix \sin(i\frac{1}{x})= x\sinh(\frac{1}{x}) \longrightarrow +\infty$$ as $x \rightarrow 0$. Therefore $f$ is not continuous at 0.