Cantor Function

The function whose values are

$${1\over 2}\left({{c_1\over 2}+\ldots+{c_{m-1}\over 2^{m-1}}+{2\over 2^m}}\right)$$

for any number between

$$a\equiv {c_1\over 3}+\ldots+{c_{m-1}\over 3^{m-1}}+{1\over 3^m}$$

and

$$b\equiv {c_1\over 3}+\ldots+{c_{m-1}\over 3^{m-1}}+{2\over 3^m}.$$

Chalice (1991) shows 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.

See also Cantor Set, Devil's Staircase

References

Chalice, D. R. ``A Characterization of the Cantor Function.'' Amer. Math. Monthly 98, 255-258, 1991.

Wagon, S. ``The Cantor Function'' and ``Complex Cantor Sets.'' §4.2 and 5.1 in Mathematica in Action. New York: W. H. Freeman, pp. 102-108 and 143-149, 1991.