Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers
The present paper is devoted to the functions from a certain subclass of non-differentiable functions. The arguments and values of the considered functions are represented by the s-adic representation or the nega-s-adic representation of real numbers. The technique of modeling these functions is the...
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irk-123456789-1458682019-02-02T01:23:18Z Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers Serbenyuk, S.O. The present paper is devoted to the functions from a certain subclass of non-differentiable functions. The arguments and values of the considered functions are represented by the s-adic representation or the nega-s-adic representation of real numbers. The technique of modeling these functions is the simplest as compared with the well-known techniques of modeling non-differentiable functions. In other words, the 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. Цю роботу присвячено деякому пiдкласу недиференцiйовних функцiй. Аргументи i значення функцiй, що розглядаються, подано через s-ве або нега-s-ве зображення дiйсних чисел. Технiка моделювання таких функцiй є простiшою в порiвняннi з добре вiдомими технiками моделювання недиференцiйовних функцiй. Iншими словами, значення цих функцiй отримано з s-го або нега-s-го зображення аргументу за допомоги певно замiни цифр чи комбiнацiй цифр. 2018 Article Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers / S.O. Serbenyuk // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 197-213. — Бібліогр.: 38 назв. — англ. 1812-9471 Mathematics Subject Classification 2010: 26A27, 11B34, 11K55, 39B22 http://dspace.nbuv.gov.ua/handle/123456789/145868 DOI: https://doi.org/10.15407/mag14.02.197 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The present paper is devoted to the functions from a certain subclass of non-differentiable functions. The arguments and values of the considered functions are represented by the s-adic representation or the nega-s-adic representation of real numbers. The technique of modeling these functions is the simplest as compared with the well-known techniques of modeling non-differentiable functions. In other words, the 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. |
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Serbenyuk, S.O. |
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Serbenyuk, S.O. Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers Журнал математической физики, анализа, геометрии |
| author_facet |
Serbenyuk, S.O. |
| author_sort |
Serbenyuk, S.O. |
| title |
Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers |
| title_short |
Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers |
| title_full |
Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers |
| title_fullStr |
Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers |
| title_full_unstemmed |
Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers |
| title_sort |
non-differentiable functions defined in terms of classical representations of real numbers |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2018 |
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http://dspace.nbuv.gov.ua/handle/123456789/145868 |
| citation_txt |
Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers / S.O. Serbenyuk // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 197-213. — Бібліогр.: 38 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT serbenyukso nondifferentiablefunctionsdefinedintermsofclassicalrepresentationsofrealnumbers |
| first_indexed |
2025-07-10T22:43:11Z |
| last_indexed |
2025-07-10T22:43:11Z |
| _version_ |
1837301833290219520 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2018, Vol. 14, No. 2, pp. 197–213
doi: https://doi.org/10.15407/mag14.02.197
Non-Differentiable Functions Defined in
Terms of Classical Representations of Real
Numbers
S.O. Serbenyuk
The present paper is devoted to the functions from a certain subclass
of non-differentiable functions. The arguments and values of the considered
functions are represented by the s-adic representation or the nega-s-adic
representation of real numbers. The technique of modeling these functions
is the simplest as compared with the well-known techniques of modeling
non-differentiable functions. In other words, the 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.
Integral, fractal and other properties of the functions are described.
Key words: nowhere differentiable function, s-adic representation, nega-
s-adic representation, non-monotonic function, Hausdorff–Besicovitch di-
mension.
Mathematical Subject Classification 2010: 26A27, 11B34, 11K55, 39B22.
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 the existence of continuous non-differentiable functions appeared
in the nineteenth century. In 1854, Dirichlet speaking at lectures at Berlin Uni-
versity said on the existence of a continuous function without derivative. In
1830, the first example of a continuous non-differentiable function was modeled
by Bolzano in “Doctrine on Function” but the paper was published one hun-
dred years later [1, 2]. In 1861, Rieman gave the following example of a non-
differentiable function without proof [37]:
f(x) =
∞∑
n=1
sin(n2x)
n2
. (1.1)
It was also studied by Hardy [3], Gerver [4], and Du Bois-Reymond. The function
has a finite derivative that equals 1
2 at the points of the form ξπ, where ξ is a
c© S.O. Serbenyuk, 2018
https://doi.org/10.15407/mag14.02.197
198 S.O. Serbenyuk
rational number with an odd numerator and an odd denominator. Function (1.1)
does not have other points of differentiability.
In 1875, Du Bois-Reymond published the following example of the function [5]:
f(x) =
∞∑
n=1
an cos(bnπx),
where 0 < a < 1 and b > 1 is an odd integer number such that ab > 1 + 3
2π. The
last-mentioned function was modeled by Weierstrass in 1871. This function has
the derivative that equals (+∞) or (−∞) on an uncountable everywhere dense
set. The following example of non-differentiable function was modeled nearly
simultaneously and independently by Darboux in the paper [6]:
f(x) =
∞∑
n=1
sin((n+ 1)!x)
n!
.
In the sequel, other examples of the 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 func-
tions 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 [13]. So the following statement is true.
Theorem 1.1 (Banach–Mazurkiewicz). The set of non-differentiable func-
tions 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 also exist functions that do not have a finite or infinite one-sided deriva-
tive at any point. In 1922, an example of such function was modeled by Besicov-
itch in [12]. The set of continuous on [0, 1] functions whose right-sided derivative
equals a finite number or equals +∞ on an uncountable set is a set of the second
Baire category in the space of all continuous functions. Hence the set of func-
tions, 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-mentioned statement was proved by Saks in 1932 (see [14]).
Now researchers are trying to find simpler examples of non-differentiable func-
tions. Interest in such functions is explained by their connection with fractals,
modeling of real objects, processes, and phenomena (in physics, economics, tech-
nology, etc.).
The present paper is devoted to the simplest examples of non-differentiable
functions defined in terms of the s-adic or nega-s-adic representations.
In addition, we consider some examples of nowhere differentiable functions
defined by other ways.
Non-Differentiable Functions 199
2. Certain examples of non-differentiable functions
Example 2.1. Consider the functions
f
(
∆3
α1α2...αn...
)
= ∆2
ϕ1(x)ϕ2(x)...ϕn(x)...
and g
(
∆s
α1α2...αn...
)
= ∆2
ϕ1(x)ϕ2(x)...ϕn(x)...
,
where s > 2 is a fixed positive integer number,
∆s
α1α2...αn... =
∞∑
n=1
αn
sn
, αn ∈ {0, 1, . . . , s− 1},
ϕ1(x) =
{
0 if α1(x) = 0
1 if α1(x) 6= 0
, ϕj(x) =
{
ϕj−1(x) for αj(x) = αj−1(x)
1− ϕj−1(x) for αj(x) 6= αj−1(x)
.
In 1952, the function g was introduced by Bush in [15], and the function f was
modeled by Wunderlich in [38]. The functions f and g are non-differentiable.
In [17], Salem modeled the function
s(x) = s
(
∆2
α1α2...αn...
)
= βα1 +
∞∑
n=2
(
βαn
n−1∏
i=1
qαi
)
= y = ∆Q2
α1α2...αn...,
where q0 > 0, q1 > 0, and q0 + q1 = 1. That is, βαn = 0 whenever αn =
0, βαn = q0 whenever αn = 1, and qαn ∈ {q0, q1}. 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 their argu-
ments are represented in terms of positive [16] or alternating [36] Cantor series or
the nega-Q̃-representation [29–31] were considered by Serbenyuk in [25–28,32,35].
Consider these generalizations of the Salem function.
Example 2.2 ([28]). Let (dn) be a fixed sequence of positive integers, dn > 1,
and (An) be a sequence of the sets An = {0, 1, . . . , dn − 1}.
Let x ∈ [0, 1] be an arbitrary number represented by a positive Cantor series
x = ∆D
ε1ε2...εn... =
∞∑
n=1
εn
d1d2 . . . dn
, where εn ∈ An.
Let P = ‖pi,n‖ be a fixed matrix such that pi,n ≥ 0, n = 1, 2, . . . , and i =
0, dn − 1,
∑dn−1
i=0 pi,n = 1 for an arbitrary n ∈ N, and
∏∞
n=1 pin,n = 0 for any
sequence (in).
Suppose that elements of the matrix P = ‖pi,n,‖ can be negative numbers as
well, but
β0,n = 0, βi,n > 0 for i 6= 0, and max
i
|pi,n| < 1.
Here
βεk,k =
{
0 if εk = 0∑εk−1
i=0 pi,k if εk 6= 0
.
Then the following statement is true.
200 S.O. Serbenyuk
Theorem 2.3. Given the matrix P such that for all n ∈ N the following
are true: pεn,npεn−1,n < 0, moreover dnpdn−1,n ≥ 1 or dnpdn−1,n ≤ 1; and the
conditions
lim
n→∞
n∏
k=1
dkp0,k 6= 0, lim
n→∞
n∏
k=1
dkpdk−1,k 6= 0
hold simultaneously. Then the function
F (x) = βε1(x),1 +
∞∑
k=2
(
βεk(x),k
k−1∏
n=1
pεn(x),n
)
is non-differentiable on [0, 1].
Example 2.4 ([35]). Let P = ‖pi,n‖ be a given matrix such that n = 1, 2, . . .
and i = 0, dn − 1. For this matrix the following system of properties holds:
1◦. ∀n ∈ N pi,n ∈ (−1, 1);
2◦. ∀n ∈ N
∑dn−1
i=0 pi,n = 1;
3◦. ∀(in), in ∈ Adn
∏∞
n=1 |pin,n| = 0;
4◦. ∀in ∈ Adn \ {0} 1 > βin,n =
∑in−1
i=0 pi,n > β0,n = 0.
Let us consider the function
F̃ (x) = βε1(x),1 +
∞∑
n=2
β̃εn(x),n n−1∏
j=1
p̃εj(x),j
,
where
β̃εn(x),n =
{
βεn(x),n if n is odd
βdn−1−εn(x),n if n is even
,
p̃εn(x),n =
{
pεn(x),n if n is odd
pdn−1−εn(x),n if n is even
,
βεn(x),n =
{
0 if εn = 0∑εn−1
i=0 pi,n if εn 6= 0
.
Here x is represented by an alternating Cantor series, i.e.,
x = ∆−(dn)ε1ε2...εn... =
∞∑
n=1
1 + εn
d1d2 . . . dn
(−1)n+1,
where (dn) is a fixed sequence of positive integers, dn > 1, and (Adn) is a sequence
of the sets Adn = {0, 1, . . . , dn − 1}, and εn ∈ Adn .
Non-Differentiable Functions 201
Theorem 2.5. Let pεn,npεn−1,n < 0 for all n ∈ N, εn ∈ Adn \ {0} and
conditions
lim
n→∞
n∏
k=1
dkp0,k 6= 0, lim
n→∞
n∏
k=1
dkpdk−1,k 6= 0
hold simultaneously. Then the function F̃ is non-differentiable on [0, 1].
Example 2.6 ([32]). Let Q̃ = ‖qi,n‖ be a fixed matrix, where i = 0,mn, mn ∈
N0
∞ = N∪{0,∞}, n = 1, 2, . . . , and the following system of properties is true for
elements qi,n of the last-mentioned matrix:
1◦. qi,n > 0;
2◦. ∀n ∈ N
∑mn
i=0 qi,n = 1;
3◦. ∀(in), in ∈ N ∪ {0}
∏∞
n=1 qin,n = 0.
The expansion of x ∈ [0, 1),
x =
i1−1∑
i=0
qi,1 +
∞∑
n=2
(−1)n−1δ̃in,n
n−1∏
j=1
q̃ij ,j
+
∞∑
n=1
2n−1∏
j=1
q̃ij ,j
, (2.1)
is called the nega-Q̃-expansion of x. By x = ∆−Q̃i1i1...in... denote the nega-Q̃-expan-
sion of x. The last-mentioned notation is called the nega-Q̃-representation of x.
Here
δ̃in,n =
1 if n is even and in = mn∑mn
i=mn−in qi,n if n is even and in 6= mn
0 if n is odd and in = 0∑in−1
i=0 qi,n if n is odd and in 6= 0
,
and the first sum in expression (2.1) is equal to 0 if i1 = 0.
Suppose that mn <∞ for all positive integers n.
Numbers from some countable subset of [0, 1] have two different nega-Q̃-
representations, i.e.,
∆−Q̃i1i2...in−1inmn+10mn+30mn+5...
= ∆−Q̃i1i2...in−1[in−1]0mn+20mn+4...
, in 6= 0.
These numbers are called nega-Q̃-rationals, and the rest of the numbers from
[0, 1] are called nega-Q̃-irrationals.
Suppose we have matrixes of the same dimension Q̃ = ‖qi,n‖ (the properties
of the last-mentioned matrix were considered earlier) and P = ‖pi,n‖, where i =
0,mn, mn ∈ N∪{0}, n = 1, 2, . . . , and for elements pi,n of P the following system
of conditions is true:
1◦. pi,n ∈ (−1, 1);
2◦. ∀n ∈ N
∑mn
i=0 pi,n = 1;
3◦. ∀(in), in ∈ N ∪ {0}
∏∞
n=1 |pin,n| = 0;
202 S.O. Serbenyuk
4◦. ∀in ∈ N 0 = β0,n < βin,n =
∑in−1
i=0 pi,n < 1.
Theorem 2.7. If the following properties of the matrix P hold:
• for all n ∈ N, in ∈ N1
mn
= {1, 2, . . . ,mn},
pin,npin−1,n < 0;
• the conditions
lim
n→∞
n∏
k=1
p0,k
q0,k
6= 0, lim
n→∞
n∏
k=1
pmk,k
qmk,k
6= 0
hold simultaneously, then the function
F (x) = βi1(x),1 +
∞∑
k=2
β̃ik(x),k k−1∏
j=1
p̃ij(x),j
does not have a finite or infinite derivative at any nega-Q̃-rational point from the
segment [0, 1].
Here
p̃in,n =
{
pin,n if n is odd
pmn−in,n if n is even
,
β̃in,n =
{
βin,n if n is odd
βmn−in,n if n is even
,
βin,n =
{∑in−1
i=0 pi,n > 0 if in 6= 0
0 if in = 0
.
The last-mentioned examples of non-differentiable functions are difficult.
However, there exist elementary examples of these functions.
3. The simplest example of non-differentiable function and its
analogues
In 2012, the main results of this subsection were represented by the author of
the present paper in [18–20,34]
We will 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]
in the following way:
x = ∆3
α1α2...αn...
f→ ∆3
ϕ(α1)ϕ(α2)...ϕ(αn)...
= f(x) = y,
where ϕ(i) = −3i2+7i
2 , i ∈ N0
2 = {0, 1, 2}, and ∆3
α1α2...αn... is the ternary repre-
sentation of x ∈ [0, 1]. That is, the values of this function are obtained from the
Non-Differentiable Functions 203
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 0.
In this subsection, differential, integral, fractal, and other properties of the
function f are described; equivalent representations of this function by addition-
ally defined auxiliary functions are considered.
We begin with the definitions of some auxiliary functions.
Let i, j, k be pairwise distinct digits of the ternary numeral system. First, let
us introduce a function ϕij(α) defined on the alphabet of the ternary numeral
system by the following:
i j k
ϕij(α) 0 0 1
That is, fij is a function given on [0, 1] in the form
x = ∆3
α1α2...αn...
fij→ ∆3
ϕij(α1)ϕij(α2)...ϕij(αn)...
= fij(x) = y.
Remark 3.1. From the definition of fij it follows that f01 = f10, f02 = f20,
and f12 = f21. Since it is true, we will use only the notations f01, f02, f12.
Lemma 3.2. The function f can be represented by:
1. f(x) = 2x− 3f01(x), where ∆3
α1α2...αn...
f01→ ∆3
ϕ01(α1)ϕ01(α2)...ϕ01(αn)...
, ϕ01(i) =
i2−i
2 , i ∈ N0
2 ;
2. f(x) = 3
2 − x − 3f12(x), where ∆3
α1α2...αn...
f12→ ∆3
ϕ12(α1)ϕ12(α2)...ϕ12(αn)...
,
ϕ12(i) = i2−3i+2
2 , i ∈ N0
2 .
3. f(x) = x
2 + 3
2f02(x), where ∆3
α1α2...αn...
f02→ ∆3
ϕ02(α1)ϕ02(α2)...ϕ02(αn)...
, ϕ02(i) =
−i2 + 2i, i ∈ N0
2 .
Lemma 3.3. The functions f, f01, f02, f12 have the properties:
1. [0, 1]
f→
(
[0, 1] \ {∆3
α1α2...αn111...}
)
∪
{
1
2
}
;
2. the point x0 = 0 is the unique invariant point of the function f ;
3. the function f is not bijective on a certain countable subset of [0, 1];
4. the following relationships hold for all x ∈ [0, 1]:
f(x)− f(1− x) = f01(x)− f12(x),
f(x) + f(1− x) =
1
2
+ 3f02(x),
f01(x) + f02(x) + f12(x) =
1
2
,
2f01(x) + f02(x) = x,
f01(x)− f12(x) = x− 1
2
;
204 S.O. Serbenyuk
5. the function f is not monotonic on the domain of definition; in particular,
the function f is a decreasing function on the set
{x : x1 < x2 ⇒ (x1 = ∆3
c1...cn01αn0+2αn0+3... ∧ x2 = ∆3
c1...cn02βn0+2βn0+3...)},
where n0 ∈ Z0 = N∪{0}, c1, c2, . . . , cn0 is an ordered set of the ternary digits,
αn0+i ∈ N0
2 , βn0+i ∈ N0
2 , i ∈ N; and the function f is an increasing function
on the set
{x : x1 < x2 ⇒ (x1 = ∆3
c1...cn00αn0+2αn0+3... ∧ x2 = ∆3
c1...cn0rβn0+2βn0+3...)},
where r ∈ {1, 2}.
Let us consider the fractal properties of all level sets of the functions
f01, f02, f12.
The set
f−1(y0) = {x : g(x) = y0},
where y0 is a fixed element of the range of values E(g) of the function g, is called
a level set of g.
Theorem 3.4. The following statements are true:
• if there exists at least one digit 2 in the ternary representation of y0, then
f−1ij (y0) = ∅;
• if y0 = 0 or y0 is a ternary-rational number from the set C[3, {0, 1}] = {y :
y = ∆3
α1α2...αn..., αn ∈ {0, 1}}, then
α0(f
−1
ij (y0)) = log3 2;
• if y0 is a ternary-irrational number from the set C[3, {0, 1}], then
0 ≤ α0(f
−1
ij (y0)) ≤ log3 2,
where α0(f
−1
ij (y0)) is the Hausdorff–Besicovitch dimension of f−1ij (y0).
Let us describe the main properties of the function f .
Theorem 3.5. 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 x0 = ∆3
α1α2...αn000... is a point of discontinuity 1
2·3n−1
whenever αn = 1, and is a point of discontinuity
(
− 1
2·3n−1
)
whenever αn = 2.
Theorem 3.6. The function f is non-differentiable.
Let us consider one fractal property of the graph of f . Suppose that
X = [0, 1]× [0, 1] =
{
(x, y) : x =
∞∑
m=1
αm
3m
, αm ∈ N0
2 , y =
∞∑
m=1
βm
3m
, βm ∈ N0
2
}
.
Non-Differentiable Functions 205
Then the set
u(α1β1)(α2β2)...(αmβm) = ∆3
α1α2...αm
×∆3
β1β2...βm
is a square with a side length of 3−m. This square is called a square of rank m
with base (α1β1)(α2β2) . . . (αmβm).
If E ⊂ X, then the number
αK(E) = inf{α : Ĥα(E) = 0} = sup{α : Ĥα(E) =∞},
where
Ĥα(E) = lim
ε→0
[
inf
d≤ε
K(E, d)dα
]
,
and K(E, d) is the minimum number of squares of the 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 αK(E) ≥ α0(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 αK(E) = α0(E) (it follows from the self-similarity
of the graph of f).
Theorem 3.7. The Hausdorff–Besicovitch dimension of the graph of f is
equal to 1.
The integral properties of f are described in the theorem below.
Theorem 3.8. The Lebesgue integral of the function f is equal to 1
2 .
There exist several analogues of the function f which have the same properties
and are defined by analogy. Let us consider these functions.
One can define 3! = 6 functions determined on [0, 1] in terms of the ternary
numeral system in the following way:
∆3
α1α2...αn...
fm→ ∆3
ϕm(α1)ϕm(α2)...ϕm(αn)...
,
where the function ϕm(αn) is determined on an alphabet of the ternary numeral
system, and fm(x) is defined by using the table for each m = 1, 6.
0 1 2
ϕ1(αn) 0 1 2
ϕ2(αn) 0 2 1
ϕ3(αn) 1 0 2
ϕ4(αn) 1 2 0
ϕ5(αn) 2 0 1
ϕ6(αn) 2 1 0
206 S.O. Serbenyuk
Thus 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 f1(x) is the function y = x and the function
f6(x) is the function y = 1− x, i.e.,
y = f1(x) = f1
(
∆3
α1α2...αn...
)
= ∆3
α1α2...αn... = x,
y = f6(x) = f6
(
∆3
α1α2...αn...
)
= ∆3
[2−α1][2−α2]...[2−αn]...
= 1− x.
We will describe some application of the function of the last-mentioned form
in the next subsection.
Lemma 3.9. Any function fm can be represented by the functions fij in the
form
fm = a(ij)m x+ b(ij)m + c(ij)m fij(x), where a(ij)m , b(ij)m , c(ij)m ∈ Q.
One can formulate the following corollary.
Theorem 3.10. The function fm such that fm(x) 6= x and fm(x) 6= 1−x is:
• continuous almost everywhere;
• non-differentiable on [0, 1];
• a function whose Hausdorff–Besicovitch dimension of the graph is equal to 1;
• a function whose Lebesgue integral is equal to 1
2 .
Generalizations of the results described in this subsection will be considered
in the following subsection.
4. Generalizations of the simplest example of non-differenti-
able function
In 2013, the investigations of the last subsection were generalized by the
author in several papers [21,22,33]. Consider these results.
We begin with the definitions.
Let s > 1 be a fixed positive integer number, and let the set A = {0, 1, . . . , s−
1} be an alphabet of the s-adic or nega-s-adic numeral system. The notation x =
∆±sα1α2...αn... means that x is represented by the s-adic or nega-s-adic representa-
tion, i.e.,
x =
∞∑
n=1
αn
sn
= ∆s
α1α2...αn...
or
x =
∞∑
n=1
(−1)nαn
sn
= ∆−sα1α2...αn..., αn ∈ A.
Non-Differentiable Functions 207
Let Λs be a class of functions of the type
f : x = ∆±sα1α2...αn... → ∆±sβ1β2...βn... = f(x) = y,
where
(
βkm+1, βkm+2, . . . , β(m+1)k
)
= θ
(
αkm+1, αkm+2, . . . , α(m+1)k
)
, the num-
ber k is a fixed positive integer for a specific function f , m = 0, 1, 2, . . . , and
θ(γ1, γ2, . . . , γk) is some function of the k variables (it is the bijective correspon-
dence) such that the set
Ak = A×A× . . .×A︸ ︷︷ ︸
k
is its domain of definition and range of values.
Each combination (γ1, γ2, . . . , γk) 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 θ(γ1, γ2, . . . , γk) of the k s-adic or nega-s-adic digits (according
to the number representation of the value of a function f). The combination
θ(γ1, γ2, . . . , γk) is assigned to the unique combination (γ
′
1, γ
′
2, . . . , γ
′
k) that may
not match with (γ1, γ2, . . . , γk). The θ is a bijective function on Ak.
It is clear that any function f ∈ Λs is one of the functions:
fsk , f+, f−1+ , f+ ◦ fsk , fsk ◦ f−1+ , f+ ◦ fsk ◦ f−1+ ,
where
fsk
(
∆s
α1α2...αn...
)
= ∆s
β1β2...βn...,(
βkm+1, βkm+2, . . . , β(m+1)k
)
= θ
(
αkm+1, αkm+2, . . . , α(m+1)k
)
for m = 0, 1, 2, . . . , and some fixed positive integer number k, i.e.,
(β1, β2, . . . , βk) = θ (α1, α2, . . . , αk) ,
(βk+1, βk+2, . . . , β2k) = θ (αk+1, αk+2, . . . , α2k) ,
. . . . . . . . . . . . . . . . . . . . .(
βkm+1, βkm+2, . . . , β(m+1)k
)
= θ
(
αkm+1, αkm+2, . . . , α(m+1)k
)
,
. . . . . . . . . . . . . . . . . . . . .
and
f+
(
∆s
α1α2...αn...
)
= ∆−sα1α2...αn...,
f−1+
(
∆−sα1α2...αn...
)
= ∆s
α1α2...αn....
Let us consider several examples.
The function f considered in the last subsection is a function of the f31 type.
In fact,
x = ∆3
α1α2...αn...
f→ ∆3
ϕ(α1)ϕ(α2)...ϕ(αn)...
= f(x) = y,
where ϕ (αn) is a function defined in terms of the s-adic numeral system in the
following way:
208 S.O. Serbenyuk
αn 0 1 2
ϕ(αn) 0 2 1
.
Now we give the example of the function f22 . The function
f22 : ∆2
α1α2...αn... → ∆2
β1β2...βn...,
where (β2m+1, β2(m+1)) = θ(α2m+1, α2(m+1)), m = 0, 1, 2, 3, . . . , and
α2m+1α2(m+1) 00 01 10 11
β2m+1β2(m+1) 10 11 00 01
,
is an example of the f22 -type function.
It is obvious that the set of f21 functions consists only of the functions y = x
and y = 1− x in the binary numeral system. But the set of f22 functions has the
order, which is equal to 4!, and includes the functions y = x and y = 1 − x as
well.
Remark 4.1. The class Λs of functions includes the following linear functions:
y = x,
y = f(x) = f
(
∆s
α1α2...αn...
)
= ∆s
[s−1−α1][s−1−α2]...[s−1−αn]...
= 1− x,
y = f(x) = f
(
∆−sα1α2...αn...
)
= ∆−s[s−1−α1][s−1−α2]...[s−1−αn]...
= −s− 1
s+ 1
− x.
These functions are called Λs-linear functions.
Remark 4.2. The last-mentioned two functions in the last remark are inter-
esting for applications in certain investigations. For example, in the case of a
positive Cantor series, the function may have the form
f(x) = f
(
∆D
ε1ε2...εn...
)
= f
( ∞∑
n=1
εn
d1d2 . . . dn
)
= ∆D
[d1−1−ε1][d2−1−ε2]...[dn−1−εn]... =
∞∑
n=1
dn − 1− εn
d1d2 . . . dn
.
It is easy to see that this function is a transformation preserving the Hausdorff–
Besicovitch dimension.
Consider the following representations by the alternating Cantor series:
∆−Dε1ε2...εn... =
∞∑
n=1
(−1)nεn
d1d2 . . . dn
,
∆−(dn)ε1ε2...εn... =
∞∑
n=1
1 + εn
d1d2 . . . dn
(−1)n+1.
In 2013, the study of the relations between positive and alternating Cantor
series, as well as other investigations of alternating Cantor series, were presented
in [23,24]. These results were later published in [36].
Consider the following results that follow from the relations between positive
and alternating Cantor series.
Non-Differentiable Functions 209
Lemma 4.3. The following functions are identity transformations:
x = ∆D
ε1ε2...εn...
f→ ∆
−(dn)
ε1[d2−1−ε2]...ε2n−1[d2n−1−ε2n]... = f(x) = y,
x = ∆−(dn)ε1ε2...εn...
g→ ∆D
ε1[d2−1−ε2]...ε2n−1[d2n−1−ε2n]... = g(x) = y.
Therefore the functions below are the DP-functions (the functions preserving
the fractal Hausdorff–Besicovitch dimension):
x = ∆D
ε1ε2...εn...
f→ ∆
−(dn)
[d1−1−ε1]ε2...[d2n−1−1−ε2n−1]ε2n...
= f(x) = y,
x = ∆−(dn)ε1ε2...εn...
g→ ∆D
[d1−1−ε1]ε2...[d2n−1−1−ε2n−1]ε2n...
= g(x) = y.
A new method for the construction of the metric, probabilistic and dimen-
sional theories for the families of representations of real numbers via studies of
special mappings (G-isomorphisms of representations), under which the symbols
of a given representation are mapped onto the same symbols of the other rep-
resentation from the same family, when these mappings preserve the Lebesgue
measure and the Hausdorff–Besicovitch dimension, follows from Remark 4.2 and
investigations of the functions f+, f
−1
+ .
Let us describe the main properties of the functions f ∈ Λs.
Lemma 4.4. For any function f from Λs, except for Λs-linear functions,
the values of the function f for different representations of s-adic rational num-
bers from [0, 1] (nega-s-adic rational numbers from [− s
s+1 ,
1
s+1 ], respectively) are
different.
Remark 4.5. From the unique representation for each s-adic irrational number
from [0, 1], it follows that the function fsk is well-defined at s-adic irrational points.
To reach that any function f ∈ Λs such that f(x) 6= x and f(x) 6= 1 − x is
well-defined on the set of s-adic rational numbers from [0, 1], we will not consider
the s-adic representation with period (s− 1).
Analogously, we will not consider the nega-s-adic representation with period
(0[s− 1]).
Lemma 4.6. The set of functions f sk with the defined operation “composition
of functions” is a finite group of order
(
sk
)
!.
Lemma 4.7. The function f ∈ Λs such that f(x) 6= x, f(x) 6= − s−1
s+1 − x,
and f(x) 6= 1− x has the following properties:
1) f reflects [0, 1] or [− s
s+1 ,
1
s+1 ] (according to the number representation of the
argument of a function f) into one of the segments [0, 1] or [− s
s+1 ,
1
s+1 ] with-
out enumerable subset of points (according to the number representation of
the value of a function f);
2) the function f is not monotonic on the domain of definition;
3) the function f is not a bijective mapping on the domain of definition.
210 S.O. Serbenyuk
Lemma 4.8. The following properties of the set of invariant points of the
function fsk are true:
• the set of invariant points of fsk is a continuum set, and its Hausdorff–Besi-
covitch dimension is equal to 1
k logs j, when there exists a set {σ1, σ2, . . . , σj}
(j ≥ 2) of k-digit combinations σ1, . . . , σj of s-adic digits such that
θ(a
(i)
1 , a
(i)
2 , . . . , a
(i)
k ) = (a
(i)
1 , a
(i)
2 , . . . , a
(i)
k ),
where σi = (a
(i)
1 a
(i)
2 . . . a
(i)
k ), i = 1, j;
• the set of invariant points of fsk is a finite set, when there exists a unique
k-digit combination σ of s-adic digits such that
θ(a1, a2, . . . , ak) = (a1, a2, . . . , ak), σ = (a1a2 . . . ak);
• the set of invariant points of fsk is an empty set, when there does not exist
any k-digit combination σ of s-adic digits such that
θ(a1, a2, . . . , ak) = (a1, a2, . . . , ak), σ = (a1a2 . . . ak).
In addition, the functions f+ and f−1+ have the following properties.
Lemma 4.9. For each x ∈ [0, 1], the function f+ satisfies the equation
f(x) + f(1− x) = −s− 1
s+ 1
.
Lemma 4.10. For each y ∈ [− s
s+1 ,
1
s+1 ], the function f−1+ satisfies the equa-
tion
f−1(y) + f−1
(
−s− 1
s+ 1
− y
)
= 1.
Lemma 4.11. 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 1
2 .
The following theorems are the main theorems about the properties of the
functions f ∈ Λs.
Theorem 4.12. A function f ∈ Λs such that f(x) 6= x, f(x) 6= − s−1
s+1 − x,
and f(x) 6= 1− x is:
• continuous at s-adic irrational or nega-s-adic irrational points, and s-adic ra-
tional 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 4.13. Let f ∈ Λs. Then the following are true:
• the Hausdorff–Besicovitch dimension of the graph of any function from the
class Λs is equal to 1;
Non-Differentiable Functions 211
•
∫
D(f)
f(x) dx =
1
2
, where D(f) is the domain of f .
So, in the present paper, 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 a nowhere differentiable function and its analogues and generalizations are
described. Equivalent representations of the considered simplest example by ad-
ditionally defined auxiliary functions were reviewed.
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Non-Differentiable Functions 213
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Received May 9, 2017, revised July 17, 2017.
S.O. Serbenyuk,
Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereschen-
kivska St., Kyiv, 01004, Ukraine,
E-mail: simon6@ukr.net
Недиференцiйовнi функцiї, визначенi в термiнах
класичних представлень дiйсних чисел
S.O. Serbenyuk
Цю роботу присвячено деякому пiдкласу недиференцiйовних фун-
кцiй. Аргументи i значення функцiй, що розглядаються, подано через
s-ве або нега-s-ве зображення дiйсних чисел. Технiка моделювання та-
ких функцiй є простiшою в порiвняннi з добре вiдомими технiками мо-
делювання недиференцiйовних функцiй. Iншими словами, значення цих
функцiй отримано з s-го або нега-s-го зображення аргументу за допо-
моги певної замiни цифр чи комбiнацiй цифр.
Описано iнтегральнi, фрактальнi та iншi властивостi розглянутих
функцiй.
Ключовi слова: нiде недиференцiйовнi функцiї, s-адичнi представ-
лення, нега-s-адичнi представлення, немонотоннi функцiї, розмiрнiсть
Гаусдорфа–Безiковича.
https://www.researchgate.net/publication/311738798
https://www.researchgate.net/publication/308273000
https://arxiv.org/abs/1602.00493
https://arxiv.org/abs/1703.02820
https://arxiv.org/abs/1703.02820
mailto:simon6@ukr.net
Introduction
Certain examples of non-differentiable functions
The simplest example of non-differentiable function and its analogues
Generalizations of the simplest example of non-differentiable function
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