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The question is as such:

If two continuous mappings $f$ and $g$ of a closed interval into itself commute, that is, $f\circ g=g\circ f$, then they do not always have a common fixed point.

-- Zorich Mathematical Analysis I

I have not managed to find a counterexample case myself. Could you help me?

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    $\begingroup$ If $f(x)=x$, then $g(x), g^2(x),\ldots$ are also fixed points of $f$ (and vice versa, of course), and these points are all distinct, and none is a fixed point of $g$. This suggests that there is no very easy construction of such functions, and the references you already found perhaps cannot be substantially improved on. $\endgroup$
    – Christian Remling
    Commented Jun 14 at 1:05
  • $\begingroup$ @ChristianRemling Thank you I will check the papers more thoroughly. $\endgroup$
    – Yinuo An
    Commented Jun 14 at 1:08
  • $\begingroup$ It says "two functions", no mention of "continuous"? $\endgroup$
    – Gerald Edgar
    Commented Jun 14 at 1:25
  • $\begingroup$ @GeraldEdgar: of course I assume continuous was meant as a hypothesis. I edited the question to improve the writing and overall presentation. $\endgroup$
    – Sam Hopkins
    Commented Jun 14 at 1:29
  • $\begingroup$ Thank you @SamHopkins $\endgroup$
    – Yinuo An
    Commented Jun 14 at 1:31

1 Answer 1

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W. M. Boyce and J. P. Huneke provided counterexamples in papers that appeared in Transactions of the American Mathematical Society in 1969.

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  • $\begingroup$ wonderful answer! $\endgroup$
    – I. Haage
    Commented Oct 9 at 14:23

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