A work problem on complex variable

Question

Let be $f:\mathbb{C}\to \mathbb{C}$ a continuous function satisfying

a) $\lim_{z\to \infty} |f(z)|=\infty$;

b) $f(\mathbb{C})$ is open.

Prove that $f(\mathbb{C})=\mathbb{C}$.

Can someone give some hint?

Answer 1

Hint

• Using the fact that $\lim_{z\to \infty }|f(z)|=\infty $, you can prove that $f(\mathbb C)$ is sequentially closed, and thus closed.

• Since $f(\mathbb C)$ is open as well, using the fact that $\mathbb C$ is connected gives you the wished result.