# MinkowskiQuestionMark

gives Minkowski's question mark function .

# Details

• Mathematical function, suitable for both symbolic and numeric manipulation.
• For a real number with continued fraction representation , the Minkowski's question mark function equals .
• For certain arguments, MinkowskiQuestionMark automatically evaluates to exact values.
• MinkowskiQuestionMark can be evaluated to arbitrary numerical precision.
• MinkowskiQuestionMark automatically threads over lists.

# Examples

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## Basic Examples(2)

Evaluate MinkowskiQuestionMark at a quadratic irrational number:

Plot MinkowskiQuestionMark over the unit interval:

## Scope(14)

### Numerical Evaluation(6)

Evaluate numerically:

Evaluate at integers:

Evaluate at rational numbers:

Evaluate at high precision:

Compute the elementwise values of an array:

Or compute the matrix MinkowskiQuestionMark function using MatrixFunction:

### Function Properties(8)

MinkowskiQuestionMark is defined for all real numbers:

Its domain is restricted to real inputs:

MinkowskiQuestionMark achieves all real values::

Hence, it is surjective:

MinkowskiQuestionMark is injective:

MinkowskiQuestionMark has no discontinuities:

However, it is singular everywhere:

MinkowskiQuestionMark is nondecreasing:

MinkowskiQuestionMark is neither non-negative or non-positive:

MinkowskiQuestionMark is neither convex nor concave:

## Applications(3)

Enumeration of rational numbers from the unit interval, based on the binary representation of the ordinal number:

First few rational numbers, according to the chosen enumeration strategy:

Plot the cumulative distribution function of so constructed rationals:

Compare it with the Minkowski's question mark function:

An unstable fixed point of the Minkowski's question mark function:

Build a continued fraction expansion of a rational number:

Compare to the builtin function:

## Properties & Relations(2)

The Minkowski's question mark function satisfies reflection identity :

For the function also satisfies :

MinkowskiQuestionMark satisfies for adjacent elements and of a Farey sequence:

Wolfram Research (2014), MinkowskiQuestionMark, Wolfram Language function, https://reference.wolfram.com/language/ref/MinkowskiQuestionMark.html.

#### Text

Wolfram Research (2014), MinkowskiQuestionMark, Wolfram Language function, https://reference.wolfram.com/language/ref/MinkowskiQuestionMark.html.

#### CMS

Wolfram Language. 2014. "MinkowskiQuestionMark." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MinkowskiQuestionMark.html.

#### APA

Wolfram Language. (2014). MinkowskiQuestionMark. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MinkowskiQuestionMark.html

#### BibTeX

@misc{reference.wolfram_2024_minkowskiquestionmark, author="Wolfram Research", title="{MinkowskiQuestionMark}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/MinkowskiQuestionMark.html}", note=[Accessed: 05-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_minkowskiquestionmark, organization={Wolfram Research}, title={MinkowskiQuestionMark}, year={2014}, url={https://reference.wolfram.com/language/ref/MinkowskiQuestionMark.html}, note=[Accessed: 05-August-2024 ]}