The Cantor function 
is the continuous but not absolutely continuous function on ![[0,1]](/images/equations/CantorFunction/Inline2.svg)
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
2004).
Chalice (1991) showed that any real-valued function 
on ![[0,1]](/images/equations/CantorFunction/Inline9.svg)
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
![I_q(n)=int_0^1[F_q(t)]^ndt
=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)](/images/equations/CantorFunction/NumberedEquation1.svg) |
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
![int_0^1[F(t)]^(F(t))dt approx 0.750387...](/images/equations/CantorFunction/NumberedEquation2.svg) |
(OEIS A113223), which seems to be just slightly greater than 3/4.
