Cantor function

In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is also referred to as the Cantor ternary function, the Lebesgue function, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase, the Cantor staircase function, and the Cantor–Lebesgue function. Georg Cantor (1884) introduced the Cantor function and mentioned that Scheeffer pointed out that it was a counterexample to an extension of the fundamental theorem of calculus claimed by Harnack. The Cantor function was discussed and popularized by Scheeffer (1884), Lebesgue (1904) and Vitali (1905).

Definition

See figure. To formally define the Cantor function c : [0,1] → [0,1], let x be in [0,1] and obtain c(x) by the following steps:

1. Express x in base 3.
2. If x contains a 1, replace every digit strictly after the first 1 by 0.
3. Replace any remaining 2s with 1s.
4. Interpret the result as a binary number. The result is c(x).

For example:

• 1/4 is 0.02020202... in base 3. There are no 1s so the next stage is still 0.02020202... This is rewritten as 0.01010101... When read in base 2, this corresponds to 1/3, so c(1/4) = 1/3.
• 1/5 is 0.01210121... in base 3. The digits after the first 1 are replaced by 0s to produce 0.01000000... This is not rewritten since there are no 2s. When read in base 2, this corresponds to 1/4, so c(1/5) = 1/4.
• 200/243 is 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. When read in base 2, this corresponds to 3/4, so c(200/243) = 3/4.

Properties

The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, ${\textstyle c(x)}$ goes from 0 to 1 as ${\textstyle x}$ goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is uniformly continuous (precisely, it is Hölder continuous of exponent α = log 2/log 3) but not absolutely continuous. It is constant on intervals of the form (0.x1x2x3...xn022222..., 0.x1x2x3...xn200000...), and every point not in the Cantor set is in one of these intervals, so its derivative is 0 outside of the Cantor set. On the other hand, it has no derivative at any point in an uncountable subset of the Cantor set containing the interval endpoints described above.

The Cantor function can also be seen as the cumulative probability distribution function of the 1/2-1/2 Bernoulli measure μ supported on the Cantor set: ${\textstyle c(x)=\mu ([0,x])}$ . This probability distribution, called the Cantor distribution, has no discrete part. That is, the corresponding measure is atomless. This is why there are no jump discontinuities in the function; any such jump would correspond to an atom in the measure.

However, no non-constant part of the Cantor function can be represented as an integral of a probability density function; integrating any putative probability density function that is not almost everywhere zero over any interval will give positive probability to some interval to which this distribution assigns probability zero. In particular, as Vitali (1905) pointed out, the function is not the integral of its derivative even though the derivative exists almost everywhere.

The Cantor function is the standard example of a singular function.

The Cantor function is non-decreasing, and so in particular its graph defines a rectifiable curve. Scheeffer (1884) showed that the arc length of its graph is 2.

Lack of absolute continuity

Because the Lebesgue measure of the uncountably infinite Cantor set is 0, for any positive ε < 1 and δ, there exists a finite sequence of pairwise disjoint sub-intervals with total length < δ over which the Cantor function cumulatively rises more than ε.

In fact, to every δ > 0 there are finitely many pairwise disjoint intervals (xk,yk) (1 ≤ k ≤ M) with $\sum \limits _{k=1}^{M}(y_{k}-x_{k})<\delta$ and $\sum \limits _{k=1}^{M}(c(y_{k})-c(x_{k}))=1$ .

Alternative definitions

Iterative construction

Below we define a sequence {ƒn} of functions on the unit interval that converges to the Cantor function.

Let ƒ0(x) = x.

Then, for every integer n ≥ 0, the next function ƒn+1(x) will be defined in terms of ƒn(x) as follows:

Let ƒn+1(x) = 0.5 × ƒn(3x),  when 0 ≤ x ≤ 1/3 ;

Let ƒn+1(x) = 0.5,  when 1/3 ≤ x ≤ 2/3 ;

Let ƒn+1(x) = 0.5 + 0.5 × ƒn(3 x − 2),  when 2/3 ≤ x ≤ 1.

The three definitions are compatible at the end-points 1/3 and 2/3, because ƒn(0) = 0 and ƒn(1) = 1 for every n, by induction. One may check that ƒn converges pointwise to the Cantor function defined above. Furthermore, the convergence is uniform. Indeed, separating into three cases, according to the definition of ƒn+1, one sees that

$\max _{x\in [0,1]}|f_{n+1}(x)-f_{n}(x)|\leq {\frac {1}{2}}\,\max _{x\in [0,1]}|f_{n}(x)-f_{n-1}(x)|,\quad n\geq 1.$ If ƒ denotes the limit function, it follows that, for every n ≥ 0,

$\max _{x\in [0,1]}|f(x)-f_{n}(x)|\leq 2^{-n+1}\,\max _{x\in [0,1]}|f_{1}(x)-f_{0}(x)|.$ Also the choice of starting function does not really matter, provided ƒ0(0) = 0, ƒ0(1) = 1 and ƒ0 is bounded[citation needed].

Fractal volume

The Cantor function is closely related to the Cantor set. The Cantor set C can be defined as the set of those numbers in the interval [0, 1] that do not contain the digit 1 in their base-3 (triadic) expansion, except if the 1 is followed by zeros only (in which case the tail 1000$\ldots$ can be replaced by 0222$\ldots$ to get rid of any 1). It turns out that the Cantor set is a fractal with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the D-dimensional volume $H_{D}$ (in the sense of a Hausdorff-measure) takes a finite value, where $D=\log(2)/\log(3)$ is the fractal dimension of C. We may define the Cantor function alternatively as the D-dimensional volume of sections of the Cantor set

$f(x)=H_{D}(C\cap (0,x)).$ Generalizations

Let

$y=\sum _{k=1}^{\infty }b_{k}2^{-k}$ be the dyadic (binary) expansion of the real number 0 ≤ y ≤ 1 in terms of binary digits bk ∈ {0,1}. Then consider the function

$C_{z}(y)=\sum _{k=1}^{\infty }b_{k}z^{k}.$ For z = 1/3, the inverse of the function x = 2 C1/3(y) is the Cantor function. That is, y = y(x) is the Cantor function. In general, for any z < 1/2, Cz(y) looks like the Cantor function turned on its side, with the width of the steps getting wider as z approaches zero.

As mentioned above, the Cantor function is also the cumulative distribution function of a measure on the Cantor set. Different Cantor functions, or Devil's Staircases, can be obtained by considering different atom-less probability measures supported on the Cantor set or other fractals. While the Cantor function has derivative 0 almost everywhere, current research focusses on the question of the size of the set of points where the upper right derivative is distinct from the lower right derivative, causing the derivative to not exist. This analysis of differentiability is usually given in terms of fractal dimension, with the Hausdorff dimension the most popular choice. This line of research was started in the 90s by Darst, who showed that the Hausdorff dimension of the set of non-differentiability of the Cantor function is the square of the dimension of the Cantor set, $(\log 2/\log 3)^{2}$ . More recently Falconer showed that this squaring relationship holds for all Ahlfor's regular, singular measures, i.e.

$\dim _{H}\left\{x:f'(x)=\lim _{h\to 0^{+}}{\frac {\mu ([x,x+h])}{h}}{\text{ does not exist}}\right\}=\left(\dim _{H}\operatorname {supp} (\mu )\right)^{2}$ Later, Troscheit obtain a more comprehensive picture of the set where the derivative does not exist for more general normalized Gibb's measures supported on self-conformal and self-similar sets.

Hermann Minkowski's question mark function loosely resembles the Cantor function visually, appearing as a "smoothed out" form of the latter; it can be constructed by passing from a continued fraction expansion to a binary expansion, just as the Cantor function can be constructed by passing from a ternary expansion to a binary expansion. The question mark function has the interesting property of having vanishing derivatives at all rational numbers.