Prove $f:[a,b]\to [a,b]$ is a homeomorphism, then $a$ and $b$ are fixed points or $f(a)=b$ & $f(b)=a$

Question

Prove $f:[a,b]\to [a,b]$ is a homeomorphism, then $a$ and $b$ are fixed points or $f(a)=b$ & $f(b)=a$

Hello. I've been struggling with this question. I found something related:

1. Continuous involutions on $\mathbb R$ with at least two fixed points. However, I am not sure how to apply it.

I know a homeomorphism is equivalent to (1) Bijection (2) $f$ continuous (3) $f$ inverse continuous. I am not sure how these conditions are related to prove the function has 2 fixed points or $f(a)=b$ and $f(b)=a$.

This is a question from my introduction to real analysis course. The only theorems proved are: (1) Brouwer fixed-point theorem (case $n$=1). (2) Banach theorem.

I’d really appreciate some help. Thanks in advance.

Answer 1

If you know a little topology: If $f(a)$ is not in $\{a,b\}$, the map $$(a,b] \to [a,b] \setminus \{f(a)\}; \quad x \mapsto f(x)$$ is a homeomorphism between a connected space and a disconnected space. Thus $f(a) \in \{a,b\}$, and similarly $f(b) \in \{a,b\}$.