TOPICS
Search

Cantor Function

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
CantorFunction

The Cantor function F(x) is a continuous function on [0,1] that is not an absolutely continuous function and may be defined as follows. First, express x in ternary. If the resulting ternary digit string contains the digit 1, replace every ternary digit following the first 1 by a 0. Next, replace all 2's with 1's. Finally, interpret the result as a binary number which then gives F(x).

The Cantor function is a particular case of a devil's staircase (Devaney 1987, p. 110), and can be extended to a function F_q for real q>2. This is the unique continuous function on [0,1] with F_q(0)=0, F_q(1)=1, and

F_q(x)={1/2F_q(qx) for 0<=x<=1/q; 1/2 for 1/q<=x<=1-1/q; 1/2+1/2F_q(qx-q+1) for 1-1/q<=x<=1,
(1)

with q=3 corresponding to the usual Cantor function (Gorin and Kukushkin 2004).

Chalice (1991) showed that any real-valued function F(x) on [0,1] which is monotone increasing and satisfies

1. F(0)=0,

2. F(x/3)=F(x)/2,

3. F(1-x)=1-F(x)

is the Cantor function (Chalice 1991; Wagon 2000, p. 132).

Gorin and Kukushkin (2004) give the remarkable identity

I_q(n)=int_0^1[F_q(t)]^ndt
(2)
=1/(n+1)-(q-2)sum_(k=1)^(|_n/2_|)(n; 2k)(2^(2k-1)-1)/(q·2^(2k-1)-1)(B_(2k))/(n-2k+1)
(3)

for nonnegative integers n, where B_(2k) is a Bernoulli number and |_x_| is the floor function. For q=3 and n=1, 2, ..., this gives the first few values as 1/2, 3/10, 1/5, 33/230, 5/46, 75/874, ... (OEIS A095844 and A095845).

Another integral associated with the Cantor function, considered by M. Trott (pers. comm., June 8, 2004), is

int_0^1[F(t)]^(F(t))dt approx 0.750372036605...
(4)

(OEIS A113223), with the integrand defined to be 1 at t=0. The value is greater than 3/4.

See also

Absolutely Continuous Function, Binary, Cantor Set, Devil's Staircase, Ternary

Explore with Wolfram|Alpha

References

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, p. 237, 2007.Chalice, D. R. "A Characterization of the Cantor Function." Amer. Math. Monthly 98, 255-258, 1991. https://doi.org/10.1080/00029890.1991.11995738.Devaney, R. L. An Introduction to Chaotic Dynamical Systems. Redwood City, CA: Addison-Wesley, 1987.Gorin, E. A. and Kukushkin, B. N. "Integrals Related to the Cantor Function." St. Petersburg Math. J. 15, 449-468, 2004. https://doi.org/10.1090/S1061-0022-04-00817-9.Sloane, N. J. A. Sequences A095844, A095845, A113223 in "The On-Line Encyclopedia of Integer Sequences."Wagon, S. "The Cantor Function" and "Complex Cantor Sets." §5.2 and 5.3 in Mathematica in Action, 2nd ed. New York: W. H. Freeman, pp. 132-138, 2000.

Referenced on Wolfram|Alpha

Cantor Function

Cite this as:

Weisstein, Eric W. "Cantor Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CantorFunction.html

Subject classifications