The Derivative of Minkowski’s ?Žx. Function

Minkowski’s ?Žx. 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 k 5.31972, and ? Žx. exists, then ? Žx. 0. In the same way, if the same average is less than k 2 log2 , where is the golden ratio, then ? Žx. . Finally some results are presented concerning metric properties of continued fractions and alternated dyadic expansions