A New Light on Minkowskiʹs ?(x) Function Original Research Article

The well-known Minkowskiʹs ?(x) function is presented as the asymptotic distribution function of an enumeration of the rationals in (0, 1] based on their continued fraction representation. The singularity of ?(x) is proved in two ways: by exhibiting a set of measure one in which ?′(x)=0; and again by actually finding a set of measure one which is mapped onto a set of measure zero and vice versa. These sets are described by means of metrical properties of different systems for real number representation.