let $f,g:\mathbb{I} \to \mathbb{I} $ continuous functions

Question

$\mathbb{I} = [0,1]$

let $f,g:\mathbb{I} \to \mathbb{I} $ continuous functions such that $ f \circ g = g \circ f$. Prove that there is $ x_0 \in \mathbb{I}$ such that $ f(x_0) =g(x_0)$.

Could you help me by giving me an idea of ​​how to do it?

Answer 1

If no such $x_0$ exists, then (WLOG) $f(x)>g(x)$ for all $x$.

Let $x_0$ be a fixed point of $f$ (I believe you can show its existence using IVT).

Then $gx_0$ is also a fixed point of $f$: $$ f(gx_0) = g(fx_0) = gx_0 $$ as are all iterates $g^Nx_0$ $$ f(g^Nx_0)=g^N(fx_0) = g^Nx_0. $$ On the other hand since $f>g$, $$ g^{N+1}x_0< g^Nx_0, $$ (they form a decreasing sequence).

By compactness, $g^n x_0 \rightarrow x_g$ for some $x_g$. This $x_g$ is a fixed point of both $g$ and $f$, a contradiction.