Properties of ?(x)
The question mark function is a strictly increasing and continuous, but not absolutely continuous function. The derivative vanishes on the rational numbers. There are several constructions for a measure that, when integrated, yields the question mark function. One such construction is obtained by measuring the density of the Farey numbers on the real number line. The question mark measure is the prototypical example of what are sometimes referred to as multi-fractal measures.
The question mark function maps rational numbers to dyadic rational numbers, meaning those whose base two representation terminates, as may be proven by induction from the recursive construction outlined above. It maps quadratic irrationals to non-dyadic rational numbers. It is an odd function, and satisfies the functional equation ?(x + 1) = ?(x) + 1; consequently x → ?(x) − x is an odd periodic function with period one. If ?(x) is irrational, then x is either algebraic of degree greater than two, or transcendental.
The graph of Minkowski question mark function is a special case of fractal curves known as de Rham curves.
Read more about this topic: Minkowski's Question Mark Function
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