Cantor Function

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, ... (OEIS A095844 and A095845).

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

(4)

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

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