The Derivative of Minkowski's ?(x) Function

Abstract

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 ḵ=2log2Φ, where Φ is the golden ratio, then ?′(x)=∞. Finally some results are presented concerning metric properties of continued fractions and alternated dyadic expansions.