Non-differentiable functions defined in terms of classical representations of real numbers

TL;DR

This paper studies a subclass of non-differentiable functions defined via s-adic and nega-s-adic representations of real numbers, offering a simpler modeling technique based on digit transformations.

Contribution

It introduces a novel, simplified method for modeling non-differentiable functions using digit modifications in s-adic and nega-s-adic representations.

Findings

Provides a new approach to model non-differentiable functions

Simplifies the analysis of functions based on number representations

Highlights the role of digit transformations in function behavior

Abstract

The present article is devoted to functions from a certain subclass of non-differentiable functions. The arguments and values of considered functions represented by the s-adic representation or the nega-s-adic representation of real numbers. The technique of modeling such functions is the simplest as compared with well-known techniques of modeling non-differentiable functions. In other words, values of these functions are obtained from the s-adic or nega-s-adic representation of the argument by a certain change of digits or combinations of digits.

Equations

$$f(x)=\sum _{n=1}^{\infty }\frac{sin(n^{2}x)}{n^2}.$$

$$f(x)=\sum _{n=1}^{\infty }{a}^{n}cos(b^n\pi x),$$

$$f(x)=\sum _{n=1}^{\infty }\frac{sin((n+1)!x)}{n!}.$$

$$f\left({\Delta }_{{\alpha }_1{\alpha }_2...{\alpha }_n...}^3\right)={\Delta }_{{\phi }_1(x){\phi }_2(x)...{\phi }_n(x)...}^2$$

$$g\left({\Delta }_{{\alpha }_1{\alpha }_2...{\alpha }_n...}^s\right)={\Delta }_{{\phi }_1(x){\phi }_2(x)...{\phi }_n(x)...}^2,$$

$${\Delta }_{{\alpha }_1{\alpha }_2...{\alpha }_n...}^s=\sum _{n=1}^{\infty }\frac{{\alpha }_n}{s^n},{\alpha }_n\in \{0,1,\dots ,s-1\},$$

$${\phi }_1(x)=\{\begin{array}{ll}0& \text{if α1(x)=0}\\ 1& \text{if α1(x)=0,}\end{array}$$

$${\phi }_j(x)=\{\begin{array}{ll}{\phi }_{j-1}(x)& \text{for αj(x)=αj−1(x)}\\ 1-{\phi }_{j-1}(x)& \text{for αj(x)=αj−1(x).}\end{array}$$

$$s(x)=s\left({\Delta }_{{\alpha }_1{\alpha }_2...{\alpha }_n...}^2\right)={\beta }_{{\alpha }_1}+\sum _{n=2}^{\infty }\left({\beta }_{{\alpha }_n}\prod _{i=1}^{n-1}{q}_i\right)=y={\Delta }_{{\alpha }_1{\alpha }_2...{\alpha }_n...}^{Q_2},$$

$$x={\Delta }_{{\epsilon }_1{\epsilon }_2...{\epsilon }_n...}^D=\sum _{n=1}^{\infty }\frac{{\epsilon }_n}{d_1{d}_2\dots d_n},\text{mbox}where{\epsilon }_n\in A_n.$$

$${\beta }_{0,n}=0,{\beta }_{i,n}>0\text{mbox}fori\ne 0,\text{mbox}and\underset{i}{\max }|p_{i,n}|<1.$$

$${\beta }_{{\epsilon }_k,k}=\{\begin{array}{ll}0& \text{if εk=0}\\ {\sum }_{i=0}^{{\epsilon }_k-1}{p}_{i,k}& \text{if εk=0.}\end{array}$$

$$\underset{n\to \infty }{\lim }\prod _{k=1}^n{d}_k{p}_{0,k}\ne 0,\underset{n\to \infty }{\lim }\prod _{k=1}^n{d}_k{p}_{d_k-1,k}\ne 0$$

$$F(x)={\beta }_{{\epsilon }_1(x),1}+\sum _{k=2}^{\infty }\left({\beta }_{{\epsilon }_k(x),k}\prod _{n=1}^{k-1}{p}_{{\epsilon }_n(x),n}\right)$$

$$\{\begin{array}{r}{1}^{\circ }.\forall n\in \mathbb{N}:p_{i,n}\in (-1,1)\\ 2^{\circ }.\forall n\in \mathbb{N}:\sum _{i=0}^{d_n-1}{p}_{i,n}=1\\ 3^{\circ }.\forall (i_n),i_n\in A_{d_n}:\prod _{n=1}^{\infty }|p_{i_n,n}|=0\\ 4^{\circ }.\forall i_n\in A_{d_n}\setminus \{0\}:1>{\beta }_{i_n,n}=\sum _{i=0}^{i_n-1}{p}_{i,n}>{\beta }_{0,n}=0.\end{array}$$

$$\tilde{F}(x)={\beta }_{{\epsilon }_1(x),1}+\sum _{n=2}^{\infty }\left({\tilde{\beta }}_{{\epsilon }_n(x),n}\prod _{j=1}^{n-1}{\tilde{p}}_{{\epsilon }_j(x),j}\right),$$

$${\tilde{\beta }}_{{\epsilon }_n(x),n}=\{\begin{array}{ll}{\beta }_{{\epsilon }_n(x),n}& \text{if n is odd }\\ {\beta }_{d_n-1-{\epsilon }_n(x),n}& \text{if n is even,}\end{array}$$

$${\tilde{p}}_{{\epsilon }_n(x),n}=\{\begin{array}{ll}{p}_{{\epsilon }_n(x),n}& \text{if n is odd }\\ p_{d_n-1-{\epsilon }_n(x),n}& \text{if n is even,}\end{array}$$

$${\beta }_{{\epsilon }_n(x),n}=\{\begin{array}{ll}0& \text{if εn=0}\\ {\sum }_{i=0}^{{\epsilon }_n-1}{p}_{i,n}& \text{if εn=0.}\end{array}$$

$$x={\Delta }_{{\epsilon }_1{\epsilon }_2...{\epsilon }_n...}^{-(d_n)}=\sum _{n=1}^{\infty }\frac{1+{\epsilon }_n}{d_1{d}_2\dots d_n}(-1{)}^{n+1},$$

$$\underset{n\to \infty }{\lim }\prod _{k=1}^n{d}_k{p}_{0,k}\ne 0,\underset{n\to \infty }{\lim }\prod _{k=1}^n{d}_k{p}_{d_k-1,k}\ne 0$$

$$\{\begin{array}{r}{1}^{\circ }.q_{i,n}>0\\ 2^{\circ }.\forall n\in \mathbb{N}:\sum _{i=0}^{m_n}{q}_{i,n}=1\\ 3^{\circ }.\forall (i_n),i_n\in \mathbb{N}\cup \{0\}:\prod _{n=1}^{\infty }{q}_{i_n,n}=0.\end{array}$$

$$x=\sum _{i=0}^{i_1-1}{q}_{i,1}+\sum _{n=2}^{\infty }\left[(-1{)}^{n-1}{\tilde{\delta }}_{i_n,n}\prod _{j=1}^{n-1}{\tilde{q}}_{i_j,j}\right]+\sum _{n=1}^{\infty }\left(\prod _{j=1}^{2n-1}{\tilde{q}}_{i_j,j}\right)$$

$${\tilde{\delta }}_{i_n,n}=\{\begin{array}{ll}1& \text{if n is even and in=mn}\\ {\sum }_{i=m_n-i_n}^{m_n}{q}_{i,n}& \text{if n is even and in=mn}\\ 0& \text{if n is odd and in=0}\\ {\sum }_{i=0}^{i_n-1}{q}_{i,n}& \text{if n is odd and in=0,}\end{array}$$

$${\Delta }_{i_1{i}_2...i_{n-1}{i}_n{m}_{n+1}0m_{n+3}0m_{n+5}...}^{-\tilde{Q}}={\Delta }_{i_1{i}_2...i_{n-1}[i_n-1]0m_{n+2}0m_{n+4}...}^{-\tilde{Q}},i_n\ne 0.$$

$$\{\begin{array}{r}{1}^{\circ }.p_{i,n}\in (-1,1)\\ 2^{\circ }.\forall n\in \mathbb{N}:\sum _{i=0}^{m_n}{p}_{i,n}=1\\ 3^{\circ }.\forall (i_n),i_n\in \mathbb{N}\cup \{0\}:\prod _{n=1}^{\infty }|p_{i_n,n}|=0\\ 4^{\circ }.\forall i_n\in \mathbb{N}:0={\beta }_{0,n}<{\beta }_{i_n,n}=\sum _{i=0}^{i_n-1}{p}_{i,n}<1.\end{array}$$

$$p_{i_n,n}\cdot p_{i_n-1,n}<0;$$

$$\underset{n\to \infty }{\lim }\prod _{k=1}^n\frac{p_{0,k}}{q_{0,k}}\ne 0,\underset{n\to \infty }{\lim }\prod _{k=1}^n\frac{p_{m_k,k}}{q_{m_k,k}}\ne 0$$

$$F(x)={\beta }_{i_1(x),1}+\sum _{k=2}^{\infty }\left[{\tilde{\beta }}_{i_k(x),k}\prod _{j=1}^{k-1}{\tilde{p}}_{i_j(x),j}\right].$$

$${\tilde{p}}_{i_n,n}=\{\begin{array}{ll}{p}_{i_n,n}& \text{if n is odd}\\ p_{m_n-i_n,n}& \text{if n is even, }\end{array}$$

Full text

Integral, fractal and other properties of considered functions are described.

1. Introduction

A nowhere differentiable function is a function whose derivative equals infinity or does not exist at each point from the domain of definition.

The idea of existence of continuous non-differentiable functions appeared in the nineteenth century. In 1854, Dirichlet speaking at lectures at Berlin University said on the existence of a continuous function without the derivative. In 1830, the first example of a continuous non-differentiable function was modeled by Bolzano in “Doctrine on Function” but the last paper was published one hundred years later [1, 2]. In 1861, Rieman gives the following example of a non-differentiable function without proof:

[TABLE]

Hardy [3], Gerver [4], and Du Bois-Reymond investigated the last-mentioned function. This function has a finite derivative that equals $\frac{1}{2}$ at points of the form $\xi\pi$, where $\xi$ is a rational number with an odd numerator and an odd denominator. Function (1) does not have other points of differentiability.

In 1875, Du Bois-Reymond published the following example of function [5]:

[TABLE]

where $0<a<1$ and $b>1$ is an odd integer number such that $ab>1+\frac{3}{2}\pi$. The last-mentioned function was modeled by Weierstrass in 1871. This function has the derivative that equals $(+\infty)$ or $(-\infty)$ on an uncountable everywhere dense set. In the paper [6], own example of non-differentiable function was modeled nearly simultaneously and independently by Darboux

[TABLE]

In the sequel, other examples of such functions were constructed and classes of non-differentiable functions were founded. The major contribution to these studies was made by the following scientists: Dini [9, p. 148–158], Darboux [7], Orlicz [8], Hankel [10, p. 61–65].

In 1929, the problem on the massiveness of the set of non-differentiable functions in the space of continuous functions was formulated by Steinhaus. In 1931, this problem was solved independently and by different ways by Banach [11] and Mazurkiewicz [12]. So the following statement is true.

Theorem 1** (Banach-Mazurkiewicz).**

The set of non-differentiable functions in the space $C[0,1]$ of functions, that are continuous on $[0,1]$, with the uniform metric is a set of the second category.

There exist also functions that do not have a finite or infinite one-sided derivative at any point. In 1922, an example of such function was modeled by Besicovitch in [13]. The set of continuous on $[0,1]$ functions whose right-sided derivative equals a finite number or equals $+\infty$ on an uncountable set is a set of the second Baire category in the space of all continuous functions. Hence the set of functions, that do not have a finite or infinite one-sided derivative at any point, is a set of the first category in the space of continuous on a segment functions. The last statement was proved by Saks in 1932 (see [14]).

Now researchers are trying to find more and more simple examples of non-differentiable functions. Interest in such functions is explained by them connection with fractals, modeling of real objects, processes, and phenomena (in physics, economics, technology, etc.).

The present article is devoted to simplest examples of non-differentiable functions defined in terms of the s-adic or nega-s-adic representations.

In addition, let us consider some examples of nowhere differentiable functions defined by another ways.

2. Certain examples of non-differentiable functions

Example 1**.**

Consider the following functions

[TABLE]

and

[TABLE]

where $s>2$ is a fixed positive integer number,

[TABLE]

[TABLE]

and

[TABLE]

In 1952, the function $g$ was introduced by Bush in [15], and the function $f$ was modeled by Wunderlich in [17]. The functions $f$ and $g$ are non-differentiable.**

In [16], Salem modeled the function

[TABLE]

where $q_{0}>0$, $q_{1}>0$, and $q_{0}+q_{1}=1$. This function is a singular function. However, generalizations of the Salem function can be non-differentiable functions or do not have the derivative on a certain set.

In October 2014, generalizations of the Salem function such that them arguments represented in terms of positive [28] or alternating [29] Cantor series or the nega-$\tilde{Q}$-representation [30, 27] were considered by Serbenyuk in the reports “Determination of a class of functions represented by Cantor series by systems of functional equations” and “Polybasic positive and alternating $\tilde{Q}$-representations and them applications to determination of functions by systems of functional equations” at the fractal analysis seminar of the Institute of Mathematics of the National Academy of Sciences of Ukraine and the National Pedagogical Dragomanov University (the list of talks available at

http://www.imath.kiev.ua/events/index.php?seminarId=21&archiv=1, the first presentation (in Ukrainian) available at https://www.researchgate.net/publication/314426236). These results were presented by the author in the papers (conference abstracts and articles) [23, 24, 25, 26] and in the presentation that available at

https://www.researchgate.net/publication/303736670.

Consider these generalizations of the Salem function.

Example 2** ([24]).**

Let $(d_{n})$ is a fixed sequence of positive integers, $d_{n}>1$, and $(A_{n})$ is a sequence of the sets $A_{n}=\{0,1,\dots,d_{n}-1\}$.

Let $x\in[0,1]$ be an arbitrary number represented by a positive Cantor series

[TABLE]

Let $P=||p_{i,n}||$ be a fixed matrix such that $p_{i,n}\geq 0$ ($n=1,2,\dots,$ and $i=~{}\overline{0,d_{n}-1}$), $\sum^{d_{n}-1}_{i=0}{p_{i,n}}=1$ for an arbitrary $n\in\mathbb{N}$, and $\prod^{\infty}_{n=1}{p_{i_{n},n}}=0$ for any sequence $(i_{n})$.

Suppose that elements of the matrix $P=||p_{i,n,}||$ can be negative numbers as well but

[TABLE]

Here

[TABLE]

Then the following statement is true.

Theorem 2**.**

Given the matrix $P$ such that for all $n\in\mathbb{N}$ the following are true: $p_{\varepsilon_{n},n}\cdot p_{\varepsilon_{n}-1,n}<0$ moreover $d_{n}\cdot~{}p_{d_{n}-1,n}\geq 1$ or $d_{n}\cdot p_{d_{n}-1,n}\leq 1$; and the conditions

[TABLE]

hold simultaneously. Then the function

[TABLE]

is non-differentiable on $[0,1]$.

Example 3** ([25]).**

Let $P=||p_{i,n}||$ be a given matrix such that $n=1,2,\dots$ and $i=\overline{0,d_{n}-1}$. For this matrix the following system of properties holds:

[TABLE]

Let us consider the following function

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

Here $x$ represented by an alternating Cantor series, i.e.,

[TABLE]

where $(d_{n})$ is a fixed sequence of positive integers, $d_{n}>1$, and $(A_{d_{n}})$ is a sequence of the sets $A_{d_{n}}=\{0,1,\dots,d_{n}-1\}$, and $\varepsilon_{n}\in A_{d_{n}}$.

Theorem 3**.**

Let $p_{\varepsilon_{n},n}\cdot p_{\varepsilon_{n}-1,n}<0$ for all $n\in\mathbb{N}$, $\varepsilon_{n}\in A_{d_{n}}\setminus\{0\}$ and conditions

[TABLE]

hold simultaneously. Then the function $\tilde{F}$ is non-differentiable on $[0,1]$.

Example 4** ([26]).**

Let $\tilde{Q}=||q_{i,n}||$ be a fixed matrix, where $i=\overline{0,m_{n}}$, $m_{n}\in N^{0}_{\infty}=\mathbb{N}\cup\{0,\infty\}$, $n=1,2,\dots$, and the following system of poperties is true for elements $q_{i,n}$ of the last-mentioned matrix

[TABLE]

The following expansion of $x\in[0,1)$

[TABLE]

is called the nega-$\tilde{Q}$-expansion of $x$. By $x=\Delta^{-\tilde{Q}}_{i_{1}i_{1}...i_{n}...}$ denote the nega-$\tilde{Q}$-expansion of $x$. The last-mentioned notation is called the nega-$\tilde{Q}$-representation of $x$. Here

[TABLE]

and the first sum in the expression (2) is equal to [math] if $i_{1}=0$.

Suppose that $m_{n}<\infty$ for all positive integers $n$.

Numbers from a some countable subset of $[0,1]$ have two different nega-$\tilde{Q}$-representations, i.e.,

[TABLE]

These numbers are called nega-$\tilde{Q}$-rationals and the rest of the numbers from $[0,1]$ are called nega-$\tilde{Q}$-irrationals.

Let we have matrixes of the same dimension $\tilde{Q}=||q_{i,n}||$ (properties of the last-mentioned matrix were considered earlier) and $P=||p_{i,n}||$, where $i=\overline{0,m_{n}}$, $m_{n}\in\mathbb{N}\cup\{0\}$, $n=1,2,\dots$, and for elements $p_{i,n}$ of $P$ the following system of conditions is true:

[TABLE]

Theorem 4**.**

If the following properties of the matrix $P$ hold:

• •

for all $n\in\mathbb{N}$, $i_{n}\in N^{1}_{m_{n}}=\{1,2,\dots,m_{n}\}$

[TABLE]

• •

the conditions

[TABLE]

hold simultaneously, then the function

[TABLE]

does not have the finite or infinite derivative at any nega-$\tilde{Q}$-rational point from the segment $[0,1]$.

Here

[TABLE]

[TABLE]

The last examples of non-differentiable functions are difficult. However, there exist elementary examples of such functions.

3. The simplest example of non-differentiable function, and its analogues

In 2012, the main results of this section were represented by the author of the present article at the International Scientific Conference “Asymptotic Methods in the Theory of Differential Equations” dedicated to 80th anniversary of M. I. Shkil [19] and published in the paper [18] (the working paper available at https://www.researchgate.net/publication/314409844). The version of the last-mentioned published paper into English available at https://arxiv.org/pdf/1703.02820.pdf.

We shall not consider numbers whose ternary representation has the period $(2)$ (without the number $1$). Let us consider a certain function $f$ defined on $[0,1]$ by the following way:

[TABLE]

where $\varphi(i)=\frac{-3i^{2}+7i}{2}$, $i\in N^{0}_{2}=\{0,1,2\}$, and $\Delta^{3}_{\alpha_{1}\alpha_{2}...\alpha_{n}...}$ is the ternary representation of $x\in[0,1]$. That is values of this function are obtained from the ternary representation of the argument by the following change of digits: 0 by 0, 1 by 2, and 2 by 1. This function preserves the ternary digit [math].

In this section, differential, integral, fractal, and other properties of the function $f$ are described; equivalent representations of this function by additionally defined auxiliary functions are considered.

We begin with definitions of some auxiliary functions.

Let $i,j,k$ be pairwise distinct digits of the ternary numeral system. First let us introduce a function $\varphi_{ij}(\alpha)$ defined on the alphabet of the ternary numeral system by the following:

[TABLE]

That is $f_{ij}$ is a function given on $[0;1]$ by the following way

[TABLE]

Remark 1*.*

From the definition of $f_{ij}$ it follows that $f_{01}=f_{10}$, $f_{02}=f_{20}$, and $f_{12}=f_{21}$. Since it is true, we shall use only the following notations: $f_{01},f_{02},f_{12}$.

Lemma 1**.**

The function $f$ can be represented by the following:

1. (1)

[TABLE]

$\varphi_{01}(i)=\frac{i^{2}-i}{2}$, $i\in N^{0}_{2}$; 2. (2)

[TABLE]

$\varphi_{12}(i)=\frac{i^{2}-3i+2}{2}$, $i\in N^{0}_{2}$. 3. (3)

[TABLE]

$\varphi_{02}(i)=-i^{2}+2i$, $i\in N^{0}_{2}$.

Lemma 2**.**

The functions $f,f_{01},f_{02},f_{12}$ have the following properties:

1. (1)

[TABLE] 2. (2)

the point $x_{0}=0$ is the unique invariant point of the function $f$; 3. (3)

the function $f$ is not bijective on a certain countable subset of $[0,1]$. 4. (4)

the following relationships hold for all $x\in[0,1]$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE] 5. (5)

the function $f$ is not monotonic on the domain of definition; in particular, the function $f$ is a decreasing function on the set

[TABLE]

where $n_{0}\in\mathbb{Z}_{0}=\mathbb{N}\cup\{0\}$, $c_{1},c_{2},\dots,c_{n_{0}}$ is an ordered set of the ternary digits, $\alpha_{n_{0}+i}\in N^{0}_{2}$, $\beta_{n_{0}+i}\in N^{0}_{2}$, $i\in\mathbb{N}$; and the function $f$ is an increasing function on the set

[TABLE]

where $r\in\{1,2\}$.

Let us consider fractal properties of all level sets of the functions $f_{01},f_{02},f_{12}$.

The following set

[TABLE]

where $y_{0}$ is a fixed element of the range of values $E(g)$ of the function $g$, is called a level set of $g$.

Theorem 5**.**

The following statements are true:

• •

if there exists at least one digit $2$ in the ternary representation of $y_{0}$, then $f^{-1}_{ij}(y_{0})=\varnothing$;

• •

if $y_{0}=0$ or $y_{0}$ is a ternary-rational number from the set $C[3,\{0,1\}]$, then

[TABLE]

• •

if $y_{0}$ is a ternary-irrational number from the set $C[3,\{0,1\}]$, then

[TABLE]

where $\alpha_{0}(f^{-1}_{ij}(y_{0}))$ is the Hausdorff-Besicovitch dimension of $f^{-1}_{ij}(y_{0})$.

Let us describe the main properties of the function $f$.

Theorem 6**.**

The function $f$ is continuous at ternary-irrational points, and ternary-rational points are points of discontinuity of the function. Furthermore, a ternary-rational point $x_{0}=\Delta^{3}_{\alpha_{1}\alpha_{2}...\alpha_{n}000...}$ is a point of discontinuity $\frac{1}{2\cdot 3^{n-1}}$ whenever $\alpha_{n}=1$, and is a point of discontinuity $\left(-\frac{1}{2\cdot 3^{n-1}}\right)$ whenever $\alpha_{n}=2$.

Theorem 7**.**

The function $f$ is non-differentiable.

Let us consider one fractal property of the graph of $f$. Suppose that

[TABLE]

Then the set

[TABLE]

is a square with a side length of $3^{-m}$. This square is called a square of rank $m$ with base $(\alpha_{1}\beta_{1})(\alpha_{2}\beta_{2})\ldots(\alpha_{m}\beta_{m})$.

If $E\subset X$, then the number

[TABLE]

where

[TABLE]

and $K(E,d)$ is the minimum number of squares of diameter $d$ required to cover the set $E$, is called * the fractal cell entropy dimension of the set E.* It is easy to see that $\alpha^{K}(E)\geq\alpha_{0}(E)$.

The notion of the fractal cell entropy dimension is used for the calculation of the Hausdorff-Besicovitch dimension of the graph of $f$, because, in the case of the function $f$, we obtain that $\alpha^{K}(E)=\alpha_{0}(E)$ (it follows from the self-similarity of the graph of $f$).

Theorem 8**.**

The Hausdorff-Besicovitch dimension of the graph of $f$ is equal to 1.

Integral properties of $f$ is described in the following theorem.

Theorem 9**.**

The Lebesgue integral of the function $f$ is equal to $\frac{1}{2}$.

There exist several analogues of the function $f$ such that have the same properties as this function and are defined by analogy. Let us consider these functions.

One can define $m=3!=6$ functions determined on $[0,1]$ in terms of the ternary numeral system by the following way:

[TABLE]

where the function $\varphi_{m}(\alpha_{n})$ determined on an alphabet of the ternary numeral system and $f_{m}(x)$ is defined by the following table for each $m=\overline{1,6}$.

[TABLE]

That is one can model a class of functions whose values are obtained from the ternary representation of the argument by a certain change of ternary digits.

It is easy to see that the function $f_{1}(x)$ is the function $y=x$ and the function $f_{6}(x)$ is the function $y=1-x$, i.e.,

[TABLE]

[TABLE]

We shall describe some application of function of the last form in the next subsection.

Lemma 3**.**

Any function $f_{m}$ can be represented by the functions $f_{ij}$ in the following form

[TABLE]

One can formulate the following corollary.

Theorem 10**.**

The function $f_{m}$ such that $f_{m}(x)\neq x$ and $f_{m}(x)\neq 1-x$ is:

• •

continuous almost everywhere;

• •

non-differentiable on $[0,1]$;

• •

a function whose the Hausdorff-Besicovitch dimension of the grapf is equal to $1$;

• •

a function whose the Lebesgue integral is equal to $\frac{1}{2}$.

Generalizations of results described in this section will be considered in the following subsection.

4. Generalizations of the simplest example of non-differentiable function

In 2013, the investigations of the last section were generalized by the author in several conference abstracts [20, 21] and in the paper [22] “One one class of functions with complicated local structure”. Consider these results.

We begin with definitions.

Let $s>1$ be a fixed positive integer number, and let the set $A=\{0,1,\dots,s-~{}1\}$ be an alphabet of the s-adic or nega-s-adic numeral system. The notation $x=\Delta^{\pm s}_{\alpha_{1}\alpha_{2}...\alpha_{n}...}$ means that $x$ is represented by the s-adic or nega-s-adic representation, i.e.,

[TABLE]

or

[TABLE]

Let $\Lambda_{s}$ be a class of functions of the type

[TABLE]

where $\left(\beta_{km+1},\beta_{km+2},\dots,\beta_{(m+1)k}\right)=\theta\left(\alpha_{km+1},\alpha_{km+2},\dots,\alpha_{(m+1)k}\right)$, the number $k$ is a fixed positive integer for a specific function $f$, $m=0,1,2,\dots,$ and $\theta(\gamma_{1},\gamma_{2},\dots,\gamma_{k})$ is some function of $k$ variables (it is the bijective correspondence) such that the set

[TABLE]

is its domain of definition and range of values.

Each combination $(\gamma_{1},\gamma_{2},\dots,\gamma_{k})$ of $k$ s-adic or nega-s-adic digits (according to the number representation of the argument of a function $f$) is assigned to the single combination $\theta(\gamma_{1},\gamma_{2},\dots,\gamma_{k})$ of $k$ s-adic or nega-s-adic digits (according to the number representation of the value of a function $f$). The combination $\theta(\gamma_{1},\gamma_{2},\dots,\gamma_{k})$ is assigned to the unique combination $(\gamma^{{}^{\prime}}_{1},\gamma^{{}^{\prime}}_{2},\dots,\gamma^{{}^{\prime}}_{k})$ that may be not to match with $(\gamma_{1},\gamma_{2},\dots,\gamma_{k})$. The $\theta$ is a bijective function on $A^{k}$.

It is clear that any function $f\in\Lambda_{s}$ is one of the following functions:

[TABLE]

where

[TABLE]

$\left(\beta_{km+1},\beta_{km+2},\dots,\beta_{(m+1)k}\right)=\theta\left(\alpha_{km+1},\alpha_{km+2},\dots,\alpha_{(m+1)k}\right)$ for $m=0,1,2,\dots,$ and some fixed positive integer number $k$, i.e.,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

Let us consider several examples.

The function $f$ considered in the last subsection is a function of the $f^{3}_{1}$ type. In fact,

[TABLE]

where $\varphi\left(\alpha_{n}\right)$ is a function defined in terms of the s-adic numeral system in the following way:

[TABLE]

Now, we present the example of the function $f^{2}_{2}$. The following function

[TABLE]

where $(\beta_{2m+1},\beta_{2(m+1)})=\theta(\alpha_{2m+1},\alpha_{2(m+1)})$, $m=0,1,2,3,\dots,$ and

[TABLE]

is an example of the $f^{2}_{2}$-type function.

It is obvious that the set of $f^{2}_{1}$ functions consists only of the functions $y=x$ and $y=1-x$ in the binary numeral system. But the set of $f^{2}_{2}$ functions has the order, that is equal to $4!$, and includes the functions $y=x$ and $y=1-x$ as well.

Remark 2*.*

The class $\Lambda_{s}$ of functions includes the following linear functions: $y=x$,

[TABLE]

[TABLE]

These functions are called $\Lambda_{s}$-linear functions.

Remark 3*.*

The last two functions in the last-mentioned remark are interesting for applications in certain investigations. For example, in the case of a positive Cantor series, such type function is of the form:

[TABLE]

It is easy to see the last function is a transformation preserving the Hausdorff-Besicovitch dimension.

Consider the following representations by alternating Cantor series:

[TABLE]

[TABLE]

In September 2013, the investigation of relations between positive and alternating Cantor series (and another investigations of alternating Cantor series) were presented by the author of the present article at the fractal analysis seminar of the Institute of Mathematics of the National Academy of Sciences of Ukraine and the National Pedagogical Dragomanov University (see paper [29], the presentation and the working paper available at https://www.researchgate.net/publication/303720347 and https://www.researchgate.net/publication/316787375 respectively).

Consider the following results that follow from relations between positive and alternating Cantor series.

Lemma 4**.**

The following functions are identity transformations:

[TABLE]

[TABLE]

Therefore the following functions are DP-functions (functions preserving the fractal Hausdorff-Besicovitch dimension):

[TABLE]

[TABLE]

A new method for the construction of metric, probabilistic and dimensional theories for families of representations of real numbers via studies of special mappings (G-isomorphisms of representations), under which symbols of a given representation are mapped into the same symbols of other representation from the same family, and they preserve the Lebesgue measure and the Hausdorff-Besicovitch dimension follows from investigations considered in this remark and investigations of functions $f_{+},f^{-1}_{+}$.

Let us describe the main properties of functions $f\in\Lambda_{s}$.

Lemma 5**.**

For any function $f$ from $\Lambda_{s}$ except for $\Lambda_{s}$-linear functions, values of function $f$ for different representations of s-adic rational numbers from  $[0,1]$ (nega-s-adic rational numbers from $[-\frac{s}{s+1},\frac{1}{s+1}]$ respectively) are different.

Remark 4*.*

From the unique representation for each s-adic irrational number from $[0,1]$ it follows that the function $f^{s}_{k}$ is well defined at s-adic irrational points.

To reach that any function $f\in\Lambda_{s}$ such that $f(x)\neq x$ and $f(x)\neq 1-x$ be well-defined on the set of s-adic rational numbers from $[0,1]$, we shall not consider the s-adic representation, which has period $(s-1)$.

Analogously, we shall not consider the nega-s-adic representation, which has period $(0[s-1])$.

Lemma 6**.**

The set of functions $f^{s}_{k}$ with the defined operation “composition of functions” is a finite group that has order equal to $\left(s^{k}\right)!$.

Lemma 7**.**

The function $f\in\Lambda_{s}$ such that $f(x)\neq x$, $f(x)\neq-\frac{s-1}{s+1}-x$, and $f(x)\neq 1-x$ has the following properties:

1. (1)

$f$* reflects $[0,1]$ or $[-\frac{s}{s+1},\frac{1}{s+1}]$ (according to the number representation of the argument of a function $f$) into one of the segments $[0,1]$ or $[-\frac{s}{s+1},\frac{1}{s+1}]$ without enumerable subset of points (according to the number representation of the value of a function $f$) .* 2. (2)

the function $f$ is not monotonic on the domain of definition; 3. (3)

the function $f$ is not a bijective mapping on the domain of definition.

Lemma 8**.**

The following properties of the set of invariant points of the function $f^{s}_{k}$ are true:

• •

the set of invariant points of $f^{s}_{k}$ is a continuum set, and its Hausdorff–Besicovitch dimension is equal to $\frac{1}{k}\log_{s}j$, when there exists the set $\{\sigma_{1},\sigma_{2},...,\sigma_{j}\}$ $(j\geq 2)$ of k-digit combinations $\sigma_{1},...,\sigma_{j}$ of s-adic digits such that

[TABLE]

• •

the set of invariant points of $f^{s}_{k}$ is a finite set, when there exists the unique k-digit combination $\sigma$ of s-adic digits such that

[TABLE]

• •

the set of invariant points of $f^{s}_{k}$ is an empty set, when there not exist any k-digit combination $\sigma$ of s-adic digits such that

[TABLE]

In addition, the functions $f_{+}$ and $f^{-1}_{+}$ have the following properties.

Lemma 9**.**

For each $x\in[0,1]$, the function $f_{+}$ satisfies the equation

[TABLE]

Lemma 10**.**

For each $y\in[-\frac{s}{s+1},\frac{1}{s+1}]$, the function $f^{-1}_{+}$ satisfies the equation

[TABLE]

Lemma 11**.**

The set of invariant points of the function $f_{+},$ as well as $f^{-1}_{+},$ is a self-similar fractal, and its Hausdorff–Besicovitch dimension is equal to $\frac{1}{2}$.

The following theorems are the main theorems about properties of functions $f\in\Lambda_{s}$.

Theorem 11**.**

A function $f\in\Lambda_{s}$ such that $f(x)\neq x$, $f(x)\neq-\frac{s-1}{s+1}-x$, and $f(x)\neq 1-x$ is:

• •

continuous at s-adic irrational or nega-s-adic irrational points, and s-adic rational or nega-s-adic rational points are points of discontinuity of this function (according to the number representation of the argument of the function $f$);

• •

a non-differentiable function.

Theorem 12**.**

Let $f\in\Lambda_{s}$. Then the following are true:

• •

the Hausdorff–Besicovitch dimension of the graph of any function from the class $\Lambda_{s}$ is equal to $1$;

• •

[TABLE]

So, in the present article, we considered historical moments of the development of the theory of non-differentiable functions, difficult and simplest examples of such functions. Integral, fractal, and other properties of the simplest example of nowhere differentiable function and its analogues and generalizations are described. Equivalent representations of the considered simplest example by additionally defined auxiliary functions were reviewed.