Nowhere differentiable solutions of a system of functional equations

Summary

We consider systems of functional equations which make it possible to treat certain nowhere differentiable functions in a unified way. Givenb ∈ ℕ\{1}, the functional equations are of the following form:

$$f\left( {\frac{{x + v}}{b}} \right) = a_v f(x) + g_v (x)forv = 0, \ldots , b - 1,$$

((F))

with given constantsa _v ∈ ℝ, |a _v | < 1, given continuous functionsg _v : [0, 1] → ℝ and solutionf:[0, 1] → ℝ.

As shown elsewhere, it is possible to characterize nowhere differentiable functions as the only continuous solutions of certain systems of type (F). On the other hand, we show here that, if theg _v are sufficiently differentiable and min{|a ₀|,...,|a _b−1|}⩾1/b, the continuous solution of (F) is either everywhere or nowhere differentiable. Moreover, we give conditions which allow to distinguish between these two possibilities. As an application, we treat several non-differentiable functions by this method.