Constructing nowhere differentiable functions from convex functions. (English) Zbl 1063.26003

Cater, F. S.

Real Anal. Exch. 28(2002-2003), No. 2, 617-621 (2003).

The author finds an easy way how to construct a continuous nowhere differentiable function from any nondecreasing convex function mapping the unit interval onto itself. The main result is the following.
Theorem. Let \((a_n)\) be a sequence of nonnegative real numbers such that \(\sum_na_n<\infty\). Let \((b_n)\) be a strictly increasing sequence of integers such that \(b_n\) divides \(b_{n+1}\) for each \(n\), and the sequence \((a_nb_n)\) does not converge to \(0\). For each index \(j\geq 0\), let \(f_j\) be a continuous function mapping the real line onto the interval \([0,1]\) such that \(f_j=0\) at each even integer and \(f_j=1\) at each odd integer. For each integer \(k\) and each index \(j\), let \(f_j\) be convex on the interval \((2k,2k+2)\). Then the continuous function \(\sum_{j=1}^{\infty}a_jf_j(b_jx)\) has a finite left or right derivative at no point.

Reviewer: Jozef Bobok (Praha)

MSC:

26A27	Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
26A24	Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A51	Convexity of real functions in one variable, generalizations

Keywords:

nowhere differentiable function; convex function