Cantor Function - Generalizations

Generalizations

Let

be the dyadic (binary) expansion of the real number 0 ≤ y ≤ 1 in terms of binary digits b_k = {0,1}. Then consider the function

For z = 1/3, the inverse of the function x = 2 C_1/3(y) is the Cantor function. That is, y = y(x) is the Cantor function. In general, for any z < 1/2, C_z(y) looks like the Cantor function turned on its side, with the width of the steps getting wider as z approaches zero.

Minkowski's question mark function visually loosely resembles the Cantor function, having the general appearance of a "smoothed out" Cantor function, and can be constructed by passing from a continued fraction expansion to a binary expansion, just as the Cantor function can be constructed by passing from a ternary expansion to a binary expansion. The question mark function has the interesting property of having vanishing derivatives at all rational numbers.