The Cantor function is defined as the function on such that for values of on the Cantor set, i.e.,

(1)

then

(2)

which is then extended to other values by noting that is monotone and has the same values on each removed endpoint (Chalice 1991).

The Cantor function is a particular case of a devil's staircase (Devaney 1987, p. 110), and can be extended to a function for , with corresponding to the usual Cantor function (Gorin and Kukushkin 2004).

Chalice (1991) showed that any real-valued function on which is monotone increasing and satisfies

1. ,

2. ,

3.

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

Gorin and Kukushkin (2004) give the remarkable identity

(3)

for integer . For and , 2, ..., this gives the first few values as 1/2, 3/10, 1/5, 33/230, 5/46, 75/874, ... (Sloane's A095844 and A095845).

M. Trott (pers. comm., June 8, 2004) has noted that

(4)

(Sloane's A113223), which seems to be just slightly greater than 3/4.

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.

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.

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.

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