Let $f,g:[0,1]\to[0,1]$ commute ($f\circ g=g\circ f$). Suppose $f$ is weakly increasing and $g$ is continuous.
Is it necessarily true that there exists $x\in[0,1]$ such that $x=f(x)=g(x)$ ?
Motivation. $g$ necessarily has a fixed point by mean value theorem. let $F$ be its set of fixed points ($F$ is closed). Then $f|_F:F\to F$ is order preserving and should have a fixed point by Tarski's theorem (which is a common fixed point of $f,g$).
Is the above argument correct? Am I missing something?
Thanks in advance.
