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

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.

### 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
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