Проектор $\Delta^O$-зображення чисел в $\Delta^E$-зображення

Abstract

In the paper we study function $f$ that takes each argument $x$ having alternating Ostrogradsky--Sierpi\'nski--Pierce series representation to a sum of Engel series with the same elements, i.e.,
$$f\left(\sum_{n=1}^\infty\frac{\left(-1\right)^{n-1}}{q_1q_2\ldots q_n}\right)=\sum_{n=1}^\infty\frac1{\left(q_1+1\right)\cdot\ldots\cdot \left(q_n+1\right)}, \,\, q_{n+1}>q_n\in\mathbb{N}.$$
We prove that set of values for function $f$ is nowhere dense set of positive Lebesgue measure. We analyze function $f$ in terms of monotonicity, continuity and differentiability on the set of irrational numbers. We prove that function $f$ is nowhere monotonic, continuous at any irrational point and non-differentiable in almost all points (in the sense of Lebesgue measure).