One class of continuous functionswith complicated local properties related\newline to Engel series

Abstract

In the paper, we construct and study the class of continuous on $[0, 1]$functions with continuum set of peculiarities(singular, nowhere monotonic, and non-differentiable functionsare among them).The representative of this class is the function $y = f(x)$defined by the Engel representation of argument:\begin{align*}x = \sum_{n=1}^\infty\frac{1}{(2+g_1)(2+g_1+g_2)\ldots(2+g_1+g_2+\ldots+g_n)}=: \Delta^E_{g_1g_2\ldots g_n\ldots},\end{align*}where $g_n = g_n(x) \in \{ 0, 1, 2, \ldots \}$, and convergent real series\[\sum_{n=0}^\infty u_n = u_0 + u_1 + \ldots + u_n + r_n = 1,\qquad\lvert u_n \rvert < 1,\ 0 < r_n < 1,\]by the following equality\[f(\Delta^E_{g_1(x)g_2(x)\ldots g_n(x)\ldots})= r_{g_1(x)} + \sum_{k=2}^\infty\biggl( r_{g_k(x)} \prod_{i=1}^{k-1} u_{g_i(x)} \biggr).\]We study local and global properties of function $f$:structural, extremal, differential, integral, and fractal properties.