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Let $f:[0,1]\rightarrow[0,1]$ and $g:[0,1]\rightarrow[0,1]$ be continuous functions satisfying $f\circ g =g\circ f$. Prove that there is a point $x\in [0,1]$, such that $f(x)=g(x)$.
I have thought two things
1) I have that for every continuous function $f:[0,1]→[0,1]$ there exists $y∈[0,1]$ such that $f(y)=y$, then $f,g, f\circ g$ have fixed points.
2)let $h(x)=f(x)-g(x)$ if there exists $a,b \in [0,1]$ such that $h(a)<0$ and $h(b)>0$ Hence, by the intermediate value theorem, $h$ must equal $0$ at some point $c \in [0,1]$.
the problem is that I can not find a, b that meets those conditions.