Cantor function

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 after the first 1 by 0.
3. Replace all 2s with 1s.
4. Interpret the result as a binary number. The result is c(x).

For example:

Properties

The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, c ( x ) goes from 0 to 1 as 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...x_n022222..., 0.x₁x₂x₃...x_n200000...), 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: c ( x ) = μ ( [ 0 , x ] ) . This 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.

Iterative construction

Below we define a sequence {ƒ_n} 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 ƒ_n+1(x) will be defined in terms of ƒ_n(x) as follows:

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

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

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

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

If ƒ denotes the limit function, it follows that, for every n ≥ 0,

Also the choice of starting function does not really matter, provided ƒ₀(0) = 0, ƒ₀(1) = 1 and ƒ₀ is bounded.

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 … can be replaced by 0222 … 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

Generalizations

Let

be the dyadic (binary) expansion of the real number 0 ≤ y ≤ 1 in terms of binary digits b_k = {0,1}. Then consider the function

For z = 1/3, the inverse of the function x = 2 C_1/3(y) is the Cantor function. That is, y = y(x) is the Cantor function. In general, for any z < 1/2, C_z(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 its support, ( log ⁡ 2 / log ⁡ 3 ) 2 . More recently Falconer showed that this squaring relationship holds for all Ahlfor's regular, singular measures, i.e.

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.