The Derivative of Minkowski's ?(x) Function

Jaume Paradís, Pelegri Viader, Lluis Bibiloni

Research output: Contribution to journalArticleResearchpeer-review

35 Citations (Scopus)


Minkowski's ?(x) function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it enables us to prove that its derivative, when it exists in a wide sense, can only attain two values: zero and infinity. It is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than k̄=5.31972, and ?′(x) exists, then ?′(x)=0. In the same way, if the same average is less than k̄=2log2Φ, where Φ is the golden ratio, then ?′(x)=∞. Finally some results are presented concerning metric properties of continued fractions and alternated dyadic expansions. © 2001 Academic Press.
Original languageEnglish
Pages (from-to)107-125
JournalJournal of Mathematical Analysis and Applications
Issue number1
Publication statusPublished - 1 Jan 2001


  • Minkowski's function; number systems; metric number theory

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