The Wikipedia article about Minkowski's question mark function says that "the derivative vanishes on the rational numbers", i.e. that $?(x)=0$ for all $x\in\mathbb Q$. However, I have not been able to find an academic paper or book which contains this result. (I have only found it stated that the derivative is zero almost everywhere, but not that the "almost everywhere" includes all the rationals.) Since I need to refer to this fact in an academic paper of my own, Wikipedia won't do. Can anyone help me with a source?
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$\begingroup$ please do not use ? for function, it easier if you use $f(x)$ or $g(x)$ like any normal person thanks $\endgroup$– terraceFeb 3, 2017 at 1:18
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7$\begingroup$ @terrace: $?(x)$ is the standard name of that specific function. $\endgroup$– user856Feb 3, 2017 at 1:19
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$\begingroup$ well then they are trolling $\endgroup$– terraceFeb 3, 2017 at 1:22
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2$\begingroup$ @terrace ?(x) is minkowski's famous question mark function. Unless Minkowski was the first recorded troll, born before trolling was even a thing; I think it's safe to rule out the conclusion they are "trolling". $\endgroup$– user335907Feb 3, 2017 at 21:58
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$\begingroup$ @Casper Also, I'd say Minkowski's seminal work introducing the ? function is probably your best bet. $\endgroup$– user335907Feb 3, 2017 at 22:00
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1 Answer
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See Extensive bibliography on the Minkowski Question Mark Function and allied topics compiled by Giedrius Alkauskas, which currently has 69 items. See also my math StackExchange list Bibliography for Singular Functions, which contains links to some of the papers that Alkauskas did not provide a link to, such as the papers by Denjoy.
(ADDED NEXT DAY) Regarding your request for a specific reference that the Minkowski $?(x)$ function has a zero derivative at each rational number, the (apparently unpublished) manuscript On the Minkowski measure by Linas Vepštas (arXiv:0810.1265, October 2008) states the following in its Abstract: One can show by classical techniques that its derivative must vanish on all rationals. As for a specific reference to a published proof, I suggest writing to some of the authors who have published in this area, such as Alkauskas, Bibiloni, Dushistova, Moshchevitin, Paradís, Viader, etc.
answered Feb 6, 2017 at 15:30
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$\begingroup$ Thanks for the links. But the proposition I am interested in is not mentioned on either list. I have read papers that seemed promising, but I still can't find it. $\endgroup$– CasperFeb 7, 2017 at 5:22
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$\begingroup$ Not sure if this helps: "Minkowski's question-mark function, whose definition for rational arguments is closely related to the Stern-Brocot tree" -- A comment from wikipedia. Reading on Stern-Brocot tree might help you. $\endgroup$ Mar 20 at 6:16
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1$\begingroup$ @Casper: See Cyriac Antony's comment here. My comment is in case you did not see this other comment (your MSE name was not cited, so I don't know if you were notified), and I have not tried to investigate what was suggested. $\endgroup$ Mar 20 at 8:33