[Math] Check if a function is entire or holomorphic

Question

Let $\;f: \mathbb{C} \rightarrow \mathbb{C}$ be the function

$$f(z) = \left\{ \begin{array}{cl} 0 & z = 0 \\ e^{-\frac{1}{z^2}} & z
\neq 0 \end{array} \right.$$

Show that $f$ is not an entire function, but holomorphic for $\mathbb{C}
\setminus \{0\}$.

I really don't get the definitions of holomorphic and entire functions (yet). Can you please tell me how this can be proved?

Answer 1

$f$ is not even continuous at $0$: $$f(i/n)=e^{n^2}\to\infty\ne 0 $$