Minkowski's Question Mark Function

Minkowski's question mark function is the strictly increasing function on defined by and

(1)

for with continued fraction , with the sum terminating for rational . It maps quadratic surds in the open interval to rational numbers in that interval in a continuous, order-preserving manner. For the nonnegative arguments outside used below, the extension is for a nonnegative integer and .

It is implemented in the Wolfram Language as MinkowskiQuestionMark[x].

On , the function satisfies the following properties (Salem 1943).

1. is a strictly increasing function.

2. If is rational, then is of the form , with and integers.

3. If is a quadratic surd, then its continued fraction is eventually periodic, and hence is rational.

4. The function is purely singular (Denjoy 1938).

The function can also be constructed as

(2)

where and are two consecutive irreducible fractions from a Farey sequence. Starting with and , insert the mediant between each adjacent pair at every stage. After stages, is defined at arguments, and the corresponding ordinates are for , 1, ..., (Salem 1943).

For positive integers and , the function satisfies the identity

(3)

A few special values include

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

where is the golden ratio.

The function satisfies the symmetry

(12)

where . Its only rational fixed points on are 0, 1/2, and 1. There are also irrational fixed points and , where (OEIS A048819) is the Minkowski-Bower constant, the smallest positive fixed point. It is conjectured that these five fixed points are the only ones on , or four modulo 1 (Gayfulin and Shulga 2020).

Values with large terms in their continued fractions cause to have a large section of repeating 0's or 9's (E. Pegg, Jr., pers. comm., Jan. 5, 2023). Some examples include

			(13)
			(14)
			(15)