Cantor function

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In mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous. It is also referred to as the "Devil's staircase".

Contents 1 Definition 2 Properties 3 Alternative definitions 3.1 Iterative construction 3.2 Fractal volume 4 Generalizations 5 References 6 External links

[edit] Definition

The Cantor function c : [0,1] → [0,1] is defined as follows:

1. Express x in base 3. If possible, use no 1s. (This makes a difference only if the expansion ends in 022222... = 100000... or 200000... = 122222...)
2. Replace the first 1 with a 2 and everything after it with 0.
3. Replace all 2s with 1s.
4. Interpret the result as a binary number. The result is c(x).

For example:

• 1/4 becomes 0.02020202... 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 is 1/3 so c(1/4) = 1/3.
• 1/5 becomes 0.01210121... base 3; the first 1 changes to a 2 followed by 0s to produce 0.02000000...; this is rewritten as 0.01000000...; when read in base 2, this is 1/4 so c(1/5) = 1/4.

(It may be much easier to understand this definition by looking at the graph below than by grasping the algorithm.)

[edit] Properties

The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, c 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 (and hence also continuous) but not absolutely continuous. It has no derivative at any member of the Cantor set; 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.

Extended to the left with value 0 and to the right with value 1, it is the cumulative probability distribution function of a random variable that is uniformly distributed on the Cantor set. This probability distribution has no discrete part, i.e., it does not concentrate positive probability at any point. It also has no part that can be represented by a 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. See Cantor distribution. The Cantor function is the standard example of a singular function.

The Cantor function is monotone increasing, and so in particular its graph defines a rectifiable curve. The arc length of the graph is 2.

[edit] Alternative definitions

[edit] 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 notice that the choice of starting function does not really matter, provided ƒ₀(0) = 0, ƒ₀(1) = 1 and ƒ₀ is bounded.

[edit] 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. 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

[edit] 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 = (2/3) 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.

Minkowski's question mark function visually loosely resembles the Cantor function, having the general appearance of a "smoothed out" Cantor function, and 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.

[edit] References

• Cantor Set and Function at cut-the-knot

[edit] External links

• Cantor Function by Douglas Rivers, the Wolfram Demonstrations Project.
• Weistein, Eric W., "Cantor Function" from MathWorld.