A Unified Approach for Nondifferentiable Functions

Let ψ(x) denote the distance between x and the nearest integer, and fix 0 < a < 1, ab > 1, where b is not necessarily an integer. Then for any sequence θn of phases, the function f(x) = ∑∞n=0anψ(bnx + θn) has no right (left) derivative at any point x and 2 + (log a/log b) is the box-counting dimension of the graph of f. The crucial step is to obtain the smallest Lipschitz class to which ƒ belongs.