# Have commuting functions a common value ?

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 posed with $g=\operatorname{id}$ (where it can easily be solved with the help of the intermediate value theorem). The lecturer said the stronger statement above is true, but he didn't know 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.

• Yes. If not then $f(x) < g(x)$ (or the reverse inequality) by the intermediate value theorem. Then $f$ maps the minimum fixed point of $g$ to a smaller fixed point. Contradiction. – George Lowther Sep 4 '11 at 17:40
• Btw, there was a question some time ago about the multidimensional generalisation of this. That is an open problem. – George Lowther Sep 4 '11 at 17:42
• Here is the link mathoverflow.net/questions/3332/… – Gjergji Zaimi Sep 4 '11 at 17:44
• Thank you all very much for the proof and the link and the quick reply. – tomasz Sep 4 '11 at 17:58
• Well known; e.g. it's problem 518 (the last of the book) of Bernard Gelbaum's Problems in Analysis. – Pietro Majer Sep 4 '11 at 22:15

Yes. The set of fixed points of $g$ is closed, nonempty, and is mapped into itself by $f$. Letting $a\le b$ be, respectively, the minimum and maximum fixed points of $g$, we have $f(a)\ge a=g(a)$ and $f(b)\le b=g(b)$. So, by the intermediate value theorem, there is an $x\in[a,b]$ with $f(x)=g(x)$.
• I see, I mixed two distinct problems: (a) common value is true for $n=1,$ unknown for $n \geq 2,$ (b) common fixed point is false for $n=1$ and we have nothing in print about $n \geq 2$, these all being on the closed unit ball in $\mathbb R^n$ so existence of fixed points for each map is automatic – Will Jagy Sep 4 '11 at 19:50
• @Will: (b) is false for $n\ge2$. Just take a counterexample for the $n=1$ case and extend to $I^n$ by making the maps fix the coordinates $x_i$ ($i=2,\ldots,n$). – George Lowther Sep 4 '11 at 19:56