In mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ... This article is about functions in mathematics. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Absolute continuity of real functions In mathematics, a real_valued function f of a real variable is absolutely continuous if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint intervals [xk, yk], k = 1, ..., n...

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.)

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. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... // Absolute continuity of real functions In mathematics, a real-valued function f of a real variable is absolutely continuous on a specified finite or infinite interval if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint... This article is about derivatives and differentiation in mathematical calculus. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ...

Alternative definition

Below we define a sequence of functions f_n on the interval that converges to the Cantor function.

Let f₀(x) = x.

Then f_n+1(x) will be defined in terms of f_n(x).

Let f_n+1(x) = 0.5 f_n(3x) when 0 ≤ x ≤ 1/3.

Let f_n+1(x) = 0.5 when 1/3 ≤ x ≤ 2/3.

Let f_n+1(x) = 0.5 + 0.5 f_n(3 (x − 2/3)) when 2/3 ≤ x ≤ 1.

Observe that f_n converges to the Cantor function. Also notice that the choice of starting function does not really matter, provided f₀(0) = 0 and f₀(1) = 1 and f₀ is bounded. In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ...

Yet another definition

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 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 The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... In mathematics, the Hausdorff dimension is an extended non-negative real number associated to any metric space. ...

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. In mathematics, a dyadic fraction or dyadic rational is a rational number that when written as a vulgar fraction has a denominator that is a power of two, i. ...

Minkowski's question mark function visually loosely resembles the Cantor function, having the general appearance of a "smoothed out" Cantor function. The question mark function has the interesting property of having vanishing derivatives at all rational numbers. Minkowski question mark function In mathematics, the Minkowski question mark function, sometimes called the slippery devils staircase, is a function, denoted ?(x), possessing various unusual fractal properties, defined by Hermann Minkowski in 1904. ...

References

• Cantor Set and Function at cut-the-knot