On one class of functions with complicated local structure

Abstract

We introduce a class $\Lambda_{s}$ of functions with complicated local structure. Any function from the class belongs to one of three specifically defined types $f^s _k$, $f_+$, and $f^{-1} _+$ or is a specifically defined composition of two or three functions of these types. Differential, integral, fractal and other properties of such functions are investigated. In particular, all functions $f$ from $\Lambda_{s}$ such that $f(x)\ne x$, $f(x)\ne -\frac{s-1}{s+1}-x$ and $f(x)\ne 1-x$ are nonmonotonic and nowhere differentiable. The Hausdorff--Besicovitch dimension of a plot of each $f \in \Lambda_s$ is equal to $1,$ and the Lebesgue integral of $f$ is equal to $\frac 1 2$. The proof of these statements for the compositions and the corresponding proofs for $f^s _k$, $f_+$, and $f^{-1} _+$ are similar.