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Can two commuting (composition of the functions satisfies commutativity) with $f\ne g$ and both $f$,$g$ increasing functions on $[0,1]$ both be discontinuous on the set of rationals?

Context: I had recently asked (and subsequently removed) a question on the cardinality of the set of commuting functions. That question turned out trivial as pointed by @ChrisEagle. The curiosity orginates in a paper by Ritt (1923) that characterizes polynomials that commute and subsequently the literature on fixed points of commuting maps. While continuity is assumed by default on such maps, and it is not hard to come up with commuting maps that are discontinuous, I was unable to come up with an example of commuting maps with countable discontinuities or prove that they cannot exist.

Jonas Linssen
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Rex Nolan
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  • I'm probably missing something but, if both $f$ and $g$ are continuous, so are their restrictions to $\mathbf Q$. – Dog_69 Jul 06 '23 at 15:38
  • Yes, that is true. The question asks about the extent to which commuting functions can be discontinuous, in particular can two non-identical functions be discontinuous on the rationals (or any dense subset of $[0,1]$) and still commute? – Rex Nolan Jul 06 '23 at 17:33

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