Minkowski's question mark function

In mathematics, the Minkowski question mark function (or the slippery devil's staircase), denoted by $?(x)$ , is a function possessing various unusual fractal properties, defined by Hermann Minkowski (1904, pages 171–172). It maps quadratic irrationals to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. In addition, it maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.

If $[a_{0};a_{1},a_{2},\ldots ]$ is the continued fraction representation of an irrational number $x$ , then