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 $f_k^s$, $f_{+}$, 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 $f_k^s$, $f_{+}$, and $f_{+}^{-1}$ are similar.

The investigation was represented in the following conferences: 1. Third Interuniversity Scientific Conference of Young Scientists on Mathematics and Physics, Kyiv, April 25-27, 2013; 2. Fifth International Conference on Analytic Number Theory and Spatial Tessellations, Kyiv, September 16-20, 2013

Cited by in corpus (9)

• Non-differentiable functions defined in terms of classical representations of real numbers
• On certain generalizations of one function and related problems (Systems of functional equations and generalizations of certain functions)
• On one class of functions with complicated local structure that the solutions of infinite systems of functional equations (On one application of infinite systems of functional equations in the functions theory)
• Certain singular distributions and fractals
• On certain maps defined by infinite sums
• On one approach to modeling numeral systems
• On one nearly everywhere continuous and nowhere differentiable function, that defined by automaton with finite memory
• On one map between singular expansions
• More on one class of fractals