Let $f,g: I \to I := [0,1]$ be continous functions satisfying $f \circ g = g \circ f$. Does 
there exist $x_0 \in I$ such that $f(x_0) = g(x_0)$ ? 

Background: In a homework the problem was possed with $g=\operatorname{id}$ (where 
it can easily be solved with help of the intermediate value theorem). The 
lecturer said the stronger statement above is true, but he didn't knew a proof. 
I googled a little around, but could only find something about the "commuting 
function problem" (existence of a common fixed point of $f$ and $g$) that is 
known to be false.