Let $\displaystyle f(z)= \begin{cases} z \cos \left( \frac{1}{z} \right) \ \ \ x \neq 0 \\

0 \ \ \ \ \ \ \ \ \ \ \ \ \ x=0

\end{cases} $ where $\displaystyle z \in \mathbb{C}$.

Decide whether f is continous at 0 and explain your answer.

I don't think $\displaystyle f(z)$ is continuous. However, I am having some trouble constructing two sequences that tend to 0 but have different limits when put into $\displaystyle f(z)$.

I think the sequences are going to be like $\displaystyle a_n=\frac{1}{2n \pi+ \frac{\pi}{2}}$ but they aren't working!