Commuting functions on the closed interval have the same value somewhere

Question

We are given two commuting continuous functions $f,g:[0,1]\to[0,1]$.
How can we prove that $f(x)=g(x)$ for some $x\in[0,1]$?

A trivial observation is that if one of the two functions is a homeomorphism this reduces to one-dimensional Brouwer fixed point theorem.

Answer 1

Let's prove the contrapositive.

Let $f,g:[0,1]\to[0,1]$ be continuous and suppose that for no $x\in[0,1]$, $f(x)=g(x)$. Without loss of generality we may assume (thanks to the intermediate value theorem) $f>g$ everywhere. It easily follows from continuity and compactness arguments that there exists a maximal $a\in[0,1]$ with $g(a)=a$, actually $a<1$ since $a=g(a)<f(a)\leq 1$.

Then $f(g(a))=f(a)$. Suppose we had $g(f(a))=f(g(a))(=f(a))$, then $f(a)$ would be a fixed point of $g$ and by definition of $a$ we'd need to have $f(a)\leq a=g(a)$ which is impossible, so $g(f(a))\neq f(g(a))$ and $f\circ g\neq g\circ f$.