The Cantor function
is the continuous but not absolutely continuous function on which may be defined as follows. First, express in ternary. If the resulting ternary
digit string contains the digit 1, replace every ternary digit following the 1 by
a 0. Next, replace all 2's with 1's. Finally, interpret the result as a binary number
which then gives .
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
Chalice (1991) showed that any real-valued function on which is monotone increasing
is the Cantor function (Chalice 1991; Wagon 2000, p. 132).
Gorin and Kukushkin (2004) give the remarkable identity
for integer .
2, ..., this gives the first few values as 1/2, 3/10, 1/5, 33/230, 5/46, 75/874,
... (OEIS A095844 and A095845).
M. Trott (pers. comm., June 8, 2004) has noted that
(OEIS 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.
Monthly98, 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,
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.