Show that $f$ has a fixed point at $\overline{B_1(0)}$.

Question

I really don't know how to start with this. Can you please help me with this?

Suppose $\displaystyle{f: \overline{B_1(0)} \rightarrow \overline{B_1(0)}}$ is continuous and $f$ is analytic in $\displaystyle{B_1(0)}$. Prove that $f(z)$ has a fixed point in $\displaystyle{\overline{B_1(0)}}$.

How to do this? Please. I would appreciate any help. Thanks a lot.

Answer 1

This is true by Brouwer's fixed-point theorem, simply because $f$ is continuous. The fact that it is analytic on $B_1(0)$ does not matter.