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?