Minkowski's question mark function is the function 
defined by Minkowski for the purpose of mapping the quadratic surds in the open
interval 
into the rational numbers of 
in a continuous, order-preserving manner. 
takes a number having continued
fraction ![x=[0;a_1,a_2,a_3,...]](/images/equations/MinkowskisQuestionMarkFunction/Inline5.svg)
to the number
 |
(1)
|
It is implemented in the Wolfram Language as MinkowskiQuestionMark[x].
The function satisfies the following properties (Salem 1943).
1. 
is strictly increasing.
2. If 
is rational, then 
is of the form 
, with 
and 
integers.
3. If 
is a quadratic surd, then the continued fraction
is periodic, and hence 
is rational.
4. The function is purely singular (Denjoy 1938).
can also be constructed as
 |
(2)
|
where 
and 
are two consecutive irreducible fractions from the Farey
sequence. At the 
th stage of this definition, 
is defined for 
values of 
, and the ordinates corresponding to these values are 
for 
,
1, ..., 
(Salem 1943).
The function satisfies the identity
 |
(3)
|
A few special values include
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
where 
is the golden ratio.
There are four fixed points (mod 1) of 
, namely 
, 1/2, 
and 
, where 
is the Minkowski-Bower
constant (Finch 2003, pp. 441-443) 
(OEIS A048819).
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
 |  |  |
(12)
|
 |  |  |
(13)
|
 |  |  |
(14)
|
Function." J. Number Th. 73, 212-227,
1998.Yakubovich, S. "The Affirmative Solution to Salem's Problem
Revisited." 31 Dec 2014.