For dimension one, it is easy to think in samples of continuous functions $f:[a,b]\rightarrow [a,b]$ with one, two, three,... fixed points. Or even, infinitely fixed points (take the idendity map). But I am intereseted in the following result that one old teacher told me:
Theorem: For any given non-empty and closed subset $C\subseteq \overline{B}_{\mathbb{R}^n}(0,r)$, there exists a continuous function $f:\overline{B}_{\mathbb{R}^n}(0,r)\rightarrow\overline{B}_{\mathbb{R}^n}(0,r)$, where $\overline{B}_{\mathbb{R}^n}(0,r)$ is the closed ball of radius $r>0$ centered at $0$, such that the set of fixed points of $f$ is exactly the set $C$.
Can anyone give me some idea or hint to prove it? My attempts have not worked sucessfully.
Thanks in advance.