Level sets of signed Takagi functions

Abstract

This paper examines level sets of functions of the form $f(x)=\sum_{n=0}^\infty \frac{r_n}{2^n}\phi(2^n x)$, where phi(x) is the distance from x to the nearest integer, and r_n equals 1 or -1 for each n. Such functions are referred to as signed Takagi functions. The case when r_n=1 for all n is the classical Takagi function, a well-known example of a continuous but nowhere differentiable function. For f of the above form, the maximum and minimum values of f are expressed in terms of the sequence {r_n}. It is then shown that almost all level sets of f are finite (with respect to Lebesgue measure on the range of f), but the set of ordinates y with an uncountably large level set is residual in the range of f. The concept of a local level set of the Takagi function, due to Lagarias and Maddock, is extended to arbitrary signed Takagi functions. It is shown that the average number of local level sets contained in a level set of f is the reciprocal of the height of the graph of f, and consequently, this average lies between 3/2 and 2.