Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers
The paper is devoted to one infinite parametric class of continuous functions with complicated local structure such that these functions are defined in terms of alternating Cantor series representation of numbers. The main attention is given to differential, integral and other properties of these fu...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1405652018-07-11T01:23:20Z Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers Serbenyuk, S.O. The paper is devoted to one infinite parametric class of continuous functions with complicated local structure such that these functions are defined in terms of alternating Cantor series representation of numbers. The main attention is given to differential, integral and other properties of these functions. Conditions of monotony and nonmonotony are found. The functional equations system such that the function from the given class of functions is a solution of the system is indicated. 2017 Article Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers / S.O. Serbenyuk // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 1. — С. 57-81. — Бібліогр.: 11 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag13.01.057 Mathematics Subject Classification 2000: 39B72, 26A27, 26A30, 11B34, 11K55 http://dspace.nbuv.gov.ua/handle/123456789/140565 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The paper is devoted to one infinite parametric class of continuous functions with complicated local structure such that these functions are defined in terms of alternating Cantor series representation of numbers. The main attention is given to differential, integral and other properties of these functions. Conditions of monotony and nonmonotony are found. The functional equations system such that the function from the given class of functions is a solution of the system is indicated. |
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Serbenyuk, S.O. |
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Serbenyuk, S.O. Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers Журнал математической физики, анализа, геометрии |
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Serbenyuk, S.O. |
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Serbenyuk, S.O. |
| title |
Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers |
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Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers |
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Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers |
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Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers |
| title_full_unstemmed |
Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers |
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continuous functions with complicated local structure defined in terms of alternating cantor series representation of numbers |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers / S.O. Serbenyuk // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 1. — С. 57-81. — Бібліогр.: 11 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
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AT serbenyukso continuousfunctionswithcomplicatedlocalstructuredefinedintermsofalternatingcantorseriesrepresentationofnumbers |
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2025-07-10T10:45:08Z |
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2025-07-10T10:45:08Z |
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1837256478140923904 |
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Journal of Mathematical Physics, Analysis, Geometry
2017, vol. 13, No. 1, pp. 57–81
Continuous Functions with Complicated Local
Structure Defined in Terms of Alternating Cantor
Series Representation of Numbers
S.O. Serbenyuk
Institute of Mathematics of the National Academy of Sciences of Ukraine
3 Tereschenkivska Str., Kyiv-4 01004, Ukraine
E-mail: simon.mathscience@imath.kiev.ua, simon6@ukr.net
Received October 22, 2015, revised May 18, 2016
The paper is devoted to one infinite parametric class of continuous func-
tions with complicated local structure such that these functions are defined
in terms of alternating Cantor series representation of numbers. The main
attention is given to differential, integral and other properties of these func-
tions. Conditions of monotony and nonmonotony are found. The functional
equations system such that the function from the given class of functions is
a solution of the system is indicated.
Key words: alternating Cantor series, functional equations system, mono-
tonic function, continuous nowhere monotonic function, singular function,
nowhere differentiable function, distribution function.
Mathematics Subject Classification 2010: 39B72, 26A27, 26A30, 11B34,
11K55.
1. Introduction
For modeling the functions with complicated local structure (singular, con-
tinuous nowhere differentiable, nowhere monotonic functions), various represen-
tations of real numbers are used widely in modern scientific researches [1, 2, 4, 9].
For example, the representations of numbers by positive and alternating series,
whose terms are reciprocal to positive integers or their products.
In the present paper, the application of the expansions of real numbers in
infinite series, whose terms are rational, to the construction of monotonic singular
and continuous nowhere monotonic functions is considered. Traditionally, the
simplest examples of these series are s-adic and nega-s-adic expansions.
c© S.O. Serbenyuk, 2017
S.O. Serbenyuk
In 1869, the first time Georg Cantor [10] considered the expansions of real
numbers from [0; 1] in the positive series
∞∑
n=1
εn
d1d2 . . . dn
, (1)
where (dn) is a fixed sequence of positive integers dn > 1 and (Adn) is a sequence
of alphabets Adn ≡ {0, 1, . . . , dn − 1}, εn ∈ Adn .
In scientific literature the last-mentioned series is called the positive Cantor
series or Cantor series. It is easy to see that the representation of real numbers
by the Cantor series is a generalization of the s-adic numeral system. So it
would be logical to assume the possibility of representation of real numbers by
the alternating Cantor series
∞∑
n=1
(−1)nεn
d1d2 . . . dn
, (2)
that [6] is a generalization of the nega-s-adic representation of real numbers.
Since 1869, the problem on necessary and sufficient conditions of the ratio-
nality of numbers defined by the positive Cantor series has remained open. The
last-mentioned problem was solved by the author of the present paper in [5].
Among the papers on finding the necessary or sufficient conditions of rationality
of numbers, represented by the positive Cantor series, we have [11]. In the paper,
A. Oppenheim studied mostly sufficient conditions of irrationality of numbers
defined not only by the positive series (1), but also (1) with positive and neg-
ative terms such that for the last-mentioned series the conditions |εi| < di − 1
for i = 1, 2, 3, . . ., and εmεn < 0 for some m > i and n > i, where i is any fixed
integer, are true.
Investigations of the present paper are the generalization of studies of M. V.
Pratsiovytyi and A.V. Kalashnikov [3]. The similar functions, whose argument
was defined by the positive Cantor series, were studied by the author of the
present paper in [7, 8].
In the present paper, the main attention is given to the study of the main
properties of functions with complicated local structure whose arguments are
represented by the alternating Cantor series
∞∑
n=1
1 + εn
d1d2 . . . dn
(−1)n+1, εn ∈ Adn , (3)
because the domain of definition of these functions is a closed interval [0; 1] and
each number from the interval can be represented by the last-mentioned series.
58 Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1
Continuous Functions with Complicated Local Structure
Before turning to the basic research results, let us consider the problem on
belonging of expansions (2) and (3) to numeral systems.
Theorem 1. Each number x ∈ [a0 − 1; a0], where
a0 =
∞∑
n=1
(−1)n+1
d1d2 . . . dn
,
can be represented by the series (2) in not more than two ways.
Theorem 2. For any x ∈ [0; 1], there exist not more than two different se-
quences (εn) such that
x =
∞∑
n=1
1 + εn
d1d2 . . . dn
(−1)n+1.
The proofs of Theorem 1 and Theorem 2 are similar, because the sequence
(dn) is fixed and it follows that a0 = const.
Definition 1.1. The defining of an arbitrary number x from [a0 − 1; a0] (or
from [0; 1]) by the expansion in the alternating Cantor series (2) (or (3)) is called
a nega-D-expansion, where D ≡ (dn), (or nega-(dn)-expansion) of the number
x and is denoted by x = ∆−D
ε1ε2...εn... (or x = ∆−(dn)
ε1ε2...εn...). The last-mentioned
notation is called the nega-D-representation (or nega-(dn)-representation) of the
number x.
Theorem 1 follows from the next two lemmas.
Lemma 1. For any x ∈ [a0− 1; a0], there exists a sequence (εn) such that the
number x can be represented by the series (2).
P r o o f. It is obvious that
a0 = max
{ ∞∑
n=1
(−1)nεn
d1d2 . . . dn
}
≡ ∆−D
0[d2−1]0[d4−1]0[d6−1]0...,
a0 − 1 = min
{ ∞∑
n=1
(−1)nεn
d1d2 . . . dn
}
≡ ∆−D
[d1−1]0[d3−1]0[d5−1]0....
Let x be an arbitrary number from (a0 − 1; a0),
−ε1
d1
−
∞∑
k=2
d2k−1 − 1
d1d2 . . . d2k−1
< x ≤ −ε1
d1
+
∞∑
k=1
d2k − 1
d1d2 . . . d2k
Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1 59
S.O. Serbenyuk
with 0 ≤ ε1 ≤ d1 − 1, and since
[a0 − 1; a0] = I0 =
d1−1⋃
i=0
[
− i
d1
−
∞∑
k=2
d2k−1 − 1
d1d2 . . . d2k−1
;− i
d1
+
∞∑
k=1
d2k − 1
d1d2 . . . d2k
]
,
it is obtained that
−
∞∑
k=2
d2k−1 − 1
d1d2 . . . d2k−1
< x +
ε1
d1
≤
∞∑
k=1
d2k − 1
d1d2 . . . d2k
.
Let x + ε1
d1
= x1. Then the following cases are obtained:
1. If the equality
x1 =
∞∑
k=1
d2k − 1
d1d2 . . . d2k
holds, then
x = ∆−D
ε1[d2−1]0[d4−1]0... or x = ∆−D
[ε1−1]0[d3−1]0[d5−1]0...;
2. If the equality is not true, then x = − ε1
d1
+ x1, where
ε2
d1d2
−
∞∑
k=2
d2k−1 − 1
d1d2 . . . d2k−1
≤ x1 <
ε2
d1d2
+
∞∑
k=2
d2k − 1
d1d2 . . . d2k
.
In the same way, let x2 = x1 − ε2
d1d2
. Then we have:
1. If the equality
x2 =
∞∑
k=2
d2k−1 − 1
d1d2 . . . d2k−1
holds, then
x = ∆−D
ε1ε2[d3−1]0[d5−1]0... or x = ∆−D
ε1[ε2−1]0[d4−1]0[d6−1]0....
2. In another case,
x = −ε1
d1
+
ε2
d1d2
+ x2, where
− ε3
d1d2d3
−
∞∑
k=3
d2k−1 − 1
d1d2 . . . d2k−1
< x2 ≤ − ε3
d1d2d3
+
∞∑
k=2
d2k − 1
d1d2 . . . d2k
, etc.
60 Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1
Continuous Functions with Complicated Local Structure
So, for the positive integer m,
−
∑
k> m+2
2
d2k−1 − 1
d1d2 . . . d2k−1
< xm − (−1)m+1εm+1
d1d2 . . . dm+1
<
∑
k> m+1
2
d2k − 1
d1d2 . . . d2k
.
Moreover, the following cases are possible:
1.
xm+1 =
∑
k> m+2
2
d2k−1−1
d1d2...d2k−1
, if m is an odd number;
∑
k> m+1
2
d2k−1
d1d2...d2k
, if m is an even number.
In this case,
x = ∆−D
ε1ε2...εm+1[dm+2−1]0[dm+4−1]0...
or
x = ∆−D
ε1...εm[εm+1−1]0[dm+3−1]0[dm+5−1]0....
2. If there does not exist m ∈ N such that the last-mentioned system is true,
then
x =
m+1∑
n=1
(−1)nεn
d1d2 . . . dn
+ xm+1.
Continuing the process indefinitely, we obtain
x = −ε1
d1
+ x1 = . . . = −ε1
d1
+
ε2
d1d2
− ε3
d1d2d3
+ . . . +
(−1)nεn
d1d2 . . . dn
+ xn = . . . .
Hence,
x =
∞∑
n=1
(−1)nεn
d1d2 . . . dn
.
Lemma 2. The numbers
x = ∆−D
ε1ε2...εm−1εmεm+1... and x
′
= ∆−D
ε1ε2...εm−1ε
′
mε
′
m+1...
,
where εm 6= ε
′
m, are equal iff one of the systems
εm+2i−1 = dm+2i−1 − 1,
εm+2i = 0 = ε
′
m+2i−1,
ε
′
m+2i = dm+2i − 1,
ε
′
m = εm − 1
or
εm+2i = dm+2i − 1,
εm+2i−1 = 0 = ε
′
m+2i,
ε
′
m+2i−1 = dm+2i−1 − 1,
ε
′
m − 1 = εm
is satisfied for all i ∈ N.
Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1 61
S.O. Serbenyuk
P r o o f. Necessity. Let εm = ε
′
m + 1. Then
0 = x− x
′
= ∆−D
ε1ε2...εm−1εmεm+1... −∆−D
ε1ε2...εm−1ε′mε
′
m+1...
=
(−1)m
d1d2 . . . dm
+
(−1)m+1(εm+1 − ε
′
m+1)
d1d2 . . . dm+1
+ . . . +
εm+i − ε
′
m+i
d1d2...dm+i
(−1)m+i + . . .
=
(−1)m
d1d2 . . . dm
(
1 +
∞∑
i=1
(−1)i(εm+i − ε
′
m+i)
dm+1dm+2 . . . dm+i
)
.
v ≡
∞∑
i=1
(−1)i(εm+i − ε
′
m+i)
dm+1dm+2 . . . dm+i
≥ −
∞∑
i=1
dm+i − 1
dm+1dm+2 . . . dm+i
= −1.
The last inequality becomes an equality only when
εm+2i = ε
′
m+2i−1 = 0 and εm+2i−1 = dm+2i−1 − 1, ε
′
m+2i = dm+2i − 1.
That is, the conditions for the first system follow from the equality x = x
′
. It
is easy to see that the conditions for the second system follow from x = x
′
under
the assumption that ε
′
m = εm + 1.
It is obvious that the sufficiency is true.
Definition 1.2. A number x ∈ [0; 1] is called a nega-(dn)-rational number if
it can be represented by
∆−(dn)
ε1ε2...εn−1εn[dn+1−1]0[dn+3−1]0[dn+5−1]... = ∆−(dn)
ε1ε2...εn−1[εn−1]0[dn+2−1]0[dn+4−1]0[dn+6−1....
The rest of the numbers from [0; 1] are called nega-(dn)-irrational numbers
and have a unique nega-(dn)-representation.
2. The Object of Research
Let P = ||pi,n|| be a given matrix such that n = 1, 2, . . . and i = 0, dn − 1.
For the matrix, the following system of the 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.
62 Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1
Continuous Functions with Complicated Local Structure
Let x = ∆−(dn)
ε1ε2...εn.... 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 an odd number,
βdn−1−εn(x),n, if n is an even number,
p̃εn(x),n =
{
pεn(x),n, if n is an odd number,
pdn−1−εn(x),n, if n is an even number.
To study other methods of defining the considered functions, we use the re-
lation between representations of real numbers by the positive Cantor series
ε1
d1
+
ε2
d1d2
+ . . . +
εn
d1d2 . . . dn
+ . . . ≡ ∆D
ε1ε2...εn...
and the alternating Cantor series
∞∑
n=1
1 + εn
d1d2 . . . dn
(−1)n+1 ≡ ∆−(dn)
ε1ε2...εn... ≡ ∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]...
≡ ε1
d1
+
d2 − 1− ε2
d1d2
+ . . . .
It follows that
F̃ (x) = F (g(x)) = F ◦ g,
where
x = ∆−(dn)
ε1ε2...εn...
g→ ∆D
ε1[d2−1−ε2]...ε2n−1[d2n−1−ε2n]... = g(x) = y,
F
( ∞∑
n=1
εn
d1d2 . . . dn
)
= βε1,1 +
∞∑
n=2
βεn,n
n−1∏
j=1
pεj ,j
.
The notion of the shift operator of real number expansion by the positive
Cantor series is useful for studying the methods of defining the function F̃ .
Definition 2.3. A mapping ϕ̂, defined by
ϕ̂(x) = ϕ̂
( ∞∑
n=1
εn
d1d2 . . . dn
)
=
∞∑
n=2
εn
d2d3 . . . dn
,
is called a shift operator of expansion of the number x = ∆D
ε1ε2...εn... by the positive
Cantor series (1).
Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1 63
S.O. Serbenyuk
That is, ϕ̂(x) = d1x− ε1(x) ≡ d1∆D
0ε2ε3.... In addition,
ϕ̂k(x) =
∞∑
n=k+1
εn
dk+1dk+2 . . . dn
≡ d1d2 . . . dk∆D
0 . . . 0︸ ︷︷ ︸
k
εk+1εk+2εk+3...
and
i
dk
+
ϕ̂k(x)
dk
= d1d2 . . . dk−1∆D
0 . . . 0︸ ︷︷ ︸
k−1
iεk+1εk+2εk+3...
= ϕ̂k−1(x), (4)
where i is a k-th digit in the D-representation (representation by the positive
Cantor series) ∆D
ε1ε2...εn... of x.
It is easy to see that the function F̃ is a unique solution of the following
infinite functional equations systems (the systems are equivalent because equality
(4) holds) in the class of determined and bounded on [0; 1] functions:
•
f
(
ĩ(x) + ϕ̂k(y)
dk
)
= β̃i(x),k + p̃i(x),k · f(ϕ̂k(y)),
where k = 1, 2, . . . , and i ∈ Adk
, i(x) is a k-th digit in the nega-(dn)-
representation ∆−(dn)
ε1ε2...εn... of x,
ĩ(x) =
{
i(x), if k is odd;
dk − 1− i(x), if k is even;
•
f(ϕ̂k(y)) = β̃εk+1(x),k+1 + p̃εk+1(x),k+1f(ϕ̂k+1(y)),
where k=0, 1, 2, . . . , and εk+1∈Adk+1
, y=∆D
ε1[d2−1−ε2]ε3...[d2n−1−ε2n]ε2n+1....
Really,
F̃ (x) = βε1(x),1 +
k∑
n=2
β̃εn(x),n
n−1∏
j=1
p̃εj(x),j
+
k∏
j=1
p̃εj(x),j
· f(ϕ̂k(y)),
where y = ∆D
ε1(x)[d2−1−ε2(x)]ε3(x)[d4−1−ε4(x)].... Using the limit transition as k →∞
in the last-mentioned equality, we get the proving proposition is true, because
the function F̃ is determined and bounded on [0; 1] and the third property of the
matrix P holds.
The main proposition of the present section is the well-posedness of definition
of the function.
64 Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1
Continuous Functions with Complicated Local Structure
Lemma 3. The values of the function y = F̃ (x) for different representations
of nega-(dn)-rational numbers from [0; 1] are equal.
P r o o f. Consider a nega-(dn)-rational number x and the following difference:
δ = F̃ (∆−(dn)
ε1...εn−1εn[dn+1−1]0[dn+3]0...)− F̃ (∆−(dn)
ε1...εn−1[εn−1]0[dn+2−1]0[dn+4−1]0...)
=
n−1∏
j=1
p̃εj ,j
[(β̃εn,n + β̃dn+1−1,n+1p̃εn,n + β̃0,n+2p̃εn,np̃dn+1−1,n+1
+β̃dn+3−1,n+3p̃εn,np̃dn+1−1,n+1p̃0,n+2 + ...)− (β̃εn−1,n + β̃0,n+1p̃εn−1,n
+β̃dn+2−1,n+2p̃εn−1,np̃0,n+1 + β̃0,n+3p̃εn−1,np̃0,n+1p̃dn+2−1,n+2 + . . .)].
If n is even, then
δ =
n−1∏
j=1
p̃εj ,j
[βdn−1−εn,n + βdn+1−1,n+1pdn−1−εn,n
+pdn−1−εn,n
∞∑
k=2
βdn+k−1,n+k
k−1∏
j=1
pdn+j−1,n+j
]−
n−1∏
j=1
p̃εj ,j
×
βdn−εn,n + β0,n+1pdn−εn,n + pdn−εn,n
∞∑
k=2
β0,n+k
k−1∏
j=1
p0,n+j
=
n−1∏
j=1
p̃εj ,j
(−pdn−εn−1,n + (1− pdn+1−1,n+1)pdn−εn−1,n
+pdn−εn−1,n
∞∑
k=2
(1− pdn+k−1,n+k)
k−1∏
j=1
pdn+j−1,n+j
) = 0.
If n is odd, then
δ = (pεn−1,n − (1− pdn+1−1,n+1)pεn−1,n
−(1− pdn+2−1,n+2)pεn−1,npdn+1−1,n+1 − . . .)
n−1∏
j=1
p̃εj ,j = 0.
Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1 65
S.O. Serbenyuk
3. Continuity and Monotonicity
Theorem 3. The function F̃ is:
• continuous;
• monotonic non-decreasing if the elements of the matrix P are non-negative,
and strictly increasing if all elements of the matrix P are positive.
P r o o f. Continuity. Let x0 = ∆−(dn)
ε1(x0)ε2(x0)...εn0 (x0)εn0+1(x0)... be an arbitrary
number from the interval [0; 1] and x = ∆−(dn)
ε1(x)ε2(x)...εn0 (x)εn0+1(x)... be a number
such that εj(x) = εj(x0) for j = 1, n0 − 1, and εn0(x) 6= εn0(x0). Consider the
difference
F̃ (x)− F̃ (x0) =
n0−1∏
j=1
p̃εj(x0),j
(
F̃ (ϕ̂n0−1(x))− F̃ (ϕ̂n0−1(x0))
)
.
So,
|F̃ (x)− F̃ (x0)| ≤
n0−1∏
j=1
|p̃εj(x0),j |
≤
(
max
j=1,n0−1
|p̃εj(x0),j |
)n0−1
→ 0 (n0 →∞).
The last-mentioned condition and limx→x0 F̃ (x) = F̃ (x0) are equivalent.
Really, the conditions x → x0 and n0 → ∞ are equivalent for the nega-
(dn)-irrational number x0. It follows that the function F̃ is continuous at each
nega-(dn)-irrational point.
Let x0 be a nega-(dn)-rational number. In this case, a continuity of the
function F̃ at nega-(dn)-rational point x0 can be proved by the notion of unilateral
borders for the cases of odd and even n0.
Monotonicity. Let the elements pi,n of the matrix P be non-negative. It is
obvious that
F̃ (0) = F̃ (∆−(dn)
0[d2−1]0[d4−1]...) = β0,1 +
∞∑
n=2
β0,n
n−1∏
j=1
p0,j
= min
x∈[0;1]
F̃ (x) = 0,
F̃ (1) = F̃ (∆−(dn)
[d1−1]0[d3−1]0...) = βd1−1,1 +
∞∑
n=2
βdn−1,n
n−1∏
j=1
pdj−1,j
= max
x∈[0;1]
F̃ (x) = 1.
66 Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1
Continuous Functions with Complicated Local Structure
Let x1 = ∆−(dn)
ε1(x1)ε2(x1)...εn(x1)... and x2 = ∆−(dn)
ε1(x2)ε2(x2)...εn(x2)... be such that
x1 < x2. It is obvious that there exists n0 such that εj(x1) = εj(x2) for all
j = 1, n0 − 1, and εn0(x1) < εn0(x2) in the case of an odd n0, or εn0(x1) >
εn0(x2) in the case of an even n0.
Thus,
F̃ (x2)− F̃ (x1) =
n0−1∏
j=1
p̃εj(x2),j
(β̃εn0(x2),n0
− β̃εn0(x1),n0
+
∞∑
m=1
β̃εn0+m(x2),n0+m
m−1∏
j=0
p̃εn0+j(x2),n0+j
−
∞∑
m=1
β̃εn0+m(x1),n0+m
m−1∏
j=0
p̃εn0+j(x1),n0+j
).
Since
κ =
∞∑
m=1
β̃εn0+m(x2),n0+m
m−1∏
j=0
p̃εn0+j(x2),n0+j
−
∞∑
m=1
β̃εn0+m(x1),n0+m
m−1∏
j=0
p̃εn0+j(x1),n0+j
≥ −
∞∑
m=1
β̃εn0+m(x1),n0+m
m−1∏
j=0
p̃εn0+j(x1),n0+j
,
where in the case of an odd n0
κ≥−pεn0 (x1),n0
(1− pdn0+1−1,n0+1 +
∞∑
m=2
(1− pdn0+m−1,n0+m)
m−1∏
j=1
pdn0+j−1,n0+j
)
= −pεn0 (x1),n0
,
for the case of an even n0 we have
κ ≥ −pdn0−1−εn0 (x1),n0
(
max
x∈[0,1]
F̃ (ϕ̂n0(x1))
)
= −pdn0−1−εn0 (x1),n0
.
Hence, if n0 is odd, then
F̃ (x2)− F̃ (x1) =
n0−1∏
j=1
p̃εj(x2),j
(β̃εn0 (x2),n0
− β̃εn0 (x1),n0
+ κ)
Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1 67
S.O. Serbenyuk
≥
n0−1∏
j=1
p̃εj(x2),j
(pεn0 (x1),n0
+pεn0 (x1)+1,n0
+ . . .+pεn0 (x2)−1,n0
−pεn0 (x1),n0
) ≥ 0.
If n0 is even, then
F̃ (x2)− F̃ (x1) =
n0−1∏
j=1
p̃εj(x2),j
· (β̃εn0 (x2),n0
− β̃εn0(x1),n0
+ κ)
=
(
n0−1∏
i=1
p̃i,εi(x2)
)
(pdn0−1−εn0 (x1),n0
+ pdn0−εn0 (x1),n0
+ . . .
+pdn0−2−εn0 (x2),n0
− pdn0−1−εn0 (x1),n0
) ≥ 0.
It is easy to see that the condition F̃ (x2) − F̃ (x1) > 0 holds if all elements
pi,n of the matrix P are positive.
Let the elements pi,n of P be non-negative.
Let η be a random variable defined by the Cantor expansion
η =
ξ1
d1
+
ξ2
d1d2
+
ξ3
d1d2d3
+ . . . +
ξk
d1d2 . . . dk
+ . . . ≡ ∆D
ξ1ξ2...ξk...,
where
ξk =
{
εk, if k is odd;
dk − 1− εk, if k is even,
and the digits ξk (k = 1, 2, 3, . . .) are random and take the values 0, 1, . . . , dk − 1
with probabilities p0,k, p1,k, . . . , pdk−1,k. That is, ξk are independent, and P{ξk =
ik} = pik,k, ik ∈ Adk
.
From the definition of the distribution function and the expressions
{η < x} = {ξ1 < ε1(x)} ∪ {ξ1 = ε1(x), ξ2 < d2 − 1− ε2(x)} ∪ . . .
∪{ξ1 = ε1(x), ξ2 = d2 − 1− ε2(x), . . . , ξ2k−1 < ε2k−1(x)}
∪{ξ1 = ε1(x), ξ2 = d2−1−ε2(x), . . . , ξ2k−1 = ε2k−1(x), ξ2k < d2k−1−ε2k(x)}∪. . . ,
P{ξ1 = ε1(x), ξ2 = d2 − 1− ε2(x), . . . , ξ2k−1 < ε2k−1(x)}
= βε2k−1(x),2k−1
2k−2∏
j=1
p̃εj(x),j
and
P{ξ1 = ε1(x), ξ2 = d2 − 1− ε2(x), . . . , ξ2k < d2k − 1− ε2k(x)}
68 Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1
Continuous Functions with Complicated Local Structure
= βd2k−1−ε2k(x),2k
2k−1∏
j=1
p̃εj(x),j ,
it is easy to see that the following proposition is a corollary of the last-mentioned
theorem.
Corollary 3.1. The distribution function F̃η of the random variable η has the
form
F̃η(x) =
0, x < 0;
βε1(x),1 +
∑∞
k=2
[
β̃εk(x),k
∏k−1
j=1 p̃εj(x),j
]
, 0 ≤ x < 1;
1, x ≥ 1,
where p̃εj(x),j
≥ 0.
4. Integral Properties
Theorem 4. The Lebesgue integral of the function F̃ can be calculated by the
formula ∫ 1
0
F̃ (x)dx =
∞∑
n=1
β̃0,n + β̃1,n + β̃2,n + ... + β̃dn−1,n
d1d2...dn
.
P r o o f. Denote y = g(x) (the function g was defined in Section 2.). Using
the definition of F̃ (and the properties of F̃ that follow from different ways of
defining the function) and the properties of the Lebesgue integral, we have
1∫
0
F̃ (x)dx =
1
d1∫
0
F (y)dy +
2
d1∫
1
d1
F (y)dy + . . . +
∫ 1
d1−1
d1
F (y)dy
=
1
d1∫
0
p0,1F (ϕ̂(y))dy +
2
d1∫
1
d1
[p0,1 + p1,1F (ϕ̂(y))] dy
+
3
d1∫
2
d1
[β2,1 + p2,1F (ϕ̂(y))] dy + . . . +
1∫
d1−1
d1
[βd1−1,1 + pd1−1,1F (ϕ̂(y))] dy.
Since y = ε1
d1
+ 1
d1
ϕ̂(y) and dy = 1
d1
d(ϕ̂(y)), then
1∫
0
F̃ (x)dx =
p0,1
d1
1∫
0
F (ϕ̂(y))d(ϕ̂(y))+β1,1y|
2
d1
1
d1
+
p1,1
d1
1∫
0
F (ϕ̂(y))d(ϕ̂(y))+β2,1y|
3
d1
2
d1
Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1 69
S.O. Serbenyuk
+
p2,1
d1
1∫
0
F (ϕ̂(y))d(ϕ̂(y)) + . . . + βd1−1,1y|1d1−1
d1
+
pd1−1,1
d1
1∫
0
F (ϕ̂(y))d(ϕ̂(y))
=
β1,1 + β2,1 + . . . + βd1−1,1
d1
+
1
d1
1∫
0
F (ϕ̂(y))d(ϕ̂(y)).
Analogously, from the relation between D-representation and nega-(dn)-representation,
it follows that
1∫
0
F (ϕ̂(y))d(ϕ̂(y)) =
1∫
d2−1
d2
p0,2F (ϕ̂2(y))d(ϕ̂(y))
+
d2−1
d2∫
d2−2
d2
[
β1,2 + p1,2F (ϕ̂2(y))
]
d(ϕ̂(y))+. . .+
1
d2∫
0
[
βd2−1,2 + pd2−1,2F (ϕ̂2(y))
]
d(ϕ̂(y)).
Since ϕ̂(y) = d2−1−ε2
d2
+ 1
d2
ϕ̂2(y) and d(ϕ̂(y)) = 1
d2
d(ϕ̂2(y)), we obtain
1∫
0
F (ϕ̂(y))d(ϕ̂(y)) =
p0,2
d2
1∫
0
F (ϕ̂2(y))d(ϕ̂2(y)) + β1,2y|
d2−1
d2
d2−2
d2
+
p1,2
d2
1∫
0
F (ϕ̂2(y))d(ϕ̂2(y)) + . . . + βd2−1,2y|
1
d2
0 +
pd2−1,2
d2
1∫
0
F (ϕ̂2(y))d(ϕ̂2(y))
=
β1,2 + β2,2 + . . . + βd2−1,2
d2
+
1
d2
1∫
0
F (ϕ̂2(y))d(ϕ̂2(y)).
So,
1∫
0
F̃ (x)dx =
β1,1 + β2,1 + . . . + βd1−1,1
d1
+
β1,2 + β2,2 + . . . + βd2−1,2
d1d2
+
1
d1d2
1∫
0
F (ϕ̂2(y))d(ϕ̂2(y)).
70 Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1
Continuous Functions with Complicated Local Structure
Analogously,
1∫
0
F̃ (x)dx =
n∑
j=1
β̃0,j + β̃1,j + β̃2,j + . . . + β̃dj−1,j
d1d2 . . . dj
+
1
d1d2 . . . dn
∫ 1
0
F (ϕ̂n(y))d(ϕ̂n(y)).
Continuing the process indefinitely, we obtain
1∫
0
F̃ (x)dx =
∞∑
n=1
β̃0,n + β̃1,n + β̃2,n + . . . + β̃dn−1,n
d1d2 . . . dn
.
5. Self-affine Properties
Theorem 5. If the elements pi,n of the matrix P are positive, then the graph
ΓF̃ of the function F̃ in the space R2 is the set
ΓF̃ =
⋃
x∈[0;1]
(x; . . . ◦ ψεn,n ◦ . . . ◦ ψε2,2 ◦ ψε1,1(x)),
where x = ∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]...,
ψin,n :
x
′
=
1
dn
x +
ωin,n
dn
;
y
′
= β̃in,n + p̃in,ny,
ωin,n =
{
in, if n is odd;
dn − 1− in, if n is even,
in ∈ Adn.
P r o o f. Since the following expressions
f(x) = βi,1 + pi,1f(ϕ̂(x)),
f
(
i + x
d1
)
= βi,1 + pi,1f(x)
Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1 71
S.O. Serbenyuk
are equivalent for x = ∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]..., it is obvious that
ψi1,1 :
x
′
=
1
d1
x +
i1
d1
;
y
′
= βi1,1 + pi1,1y.
Consider the affine transformations ψi,2, i = 0, d2 − 1. Since the expressions
f(ϕ̂(x)) = βd2−1−i,2 + pd2−1−i,2f(ϕ̂2(x)),
f
(
d2 − 1− i + ϕ̂(x)
d2
)
= βd2−1−i,2 + pd2−1−i,2f(ϕ̂(x))
are equivalent, we have
ψi2,2 :
x
′
=
1
d2
x +
d2 − 1− i2
d2
;
y
′
= βd2−1−i2,2 + pd2−1−i2,2y.
By induction, we obtain
ψin,n :
x
′
=
1
dn
x +
ωin,n
dn
;
y
′
= β̃in,n + p̃in,ny.
So, ⋃
x∈[0;1]
(x; . . . ◦ ψεn,n ◦ . . . ◦ ψε2,2 ◦ ψε1,1(x)) ≡ G ⊂ ΓF̃ .
Let T (x0, F̃ (x0)) ∈ ΓF̃ . Consider a point xn = ϕ̂n(x0), where x0 =
∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]... is a fixed point from [0; 1].
Since for any n ∈ N, εn and dn − 1− εn belong to Adn ,
f
(
ϕ̂k(x0)
)
= β̃εk+1,k+1 + p̃εk+1,k+1f
(
ϕ̂k+1(x0)
)
, k = 0, 1, . . .
and from T
(
ϕ̂k(x0); F̃
(
ϕ̂k(x0)
)) ∈ ΓF̃ , it follows that
ψik,k ◦ . . . ◦ ψi2,2 ◦ ψi1,1
(
T
)
= T0(x0; F̃ (x0)) ∈ ΓF̃ , ik ∈ Adk
, k →∞.
Therefore, ΓF̃ ⊂ G, and thus
ΓF̃ =
⋃
x∈[0;1]
(x; . . . ◦ ψεn,n ◦ . . . ◦ ψε2,2 ◦ ψε1,1(x)).
72 Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1
Continuous Functions with Complicated Local Structure
6. Differential Properties when the Elements of the Matrix P
are Non-negative
Let the elements pi,n of the matrix P be non-negative.
Definition 6.4. Let c1, c2, . . . , cn be an ordered set of integer numbers such
that ci ∈ Adi for all i = 1, n.
A cylinder ∆−(dn)
c1c2...cn of rank n with the base c1c2 . . . cn is called a set of all
numbers from [0; 1] such that the first n digits of the nega-(dn)-representation of
the numbers are equal to c1, c2, . . . , cn. That is,
∆−(dn)
c1c2...cn
≡
{
x : x = ∆−(dn)
c1c2...cnεn+1...εn+k..., εn+k ∈ Adn+k
}
.
Definition 6.5. The change µF̃ in the function F̃ on the cylinder ∆−(dn)
c1c2...cn
is called a value µF̃
(
∆−(dn)
c1c2...cn
)
defined by the equality
µF̃
(
∆−(dn)
c1c2...cn
)
= F̃
(
sup∆−(dn)
c1c2...cn
)
− F̃
(
inf ∆−(dn)
c1c2...cn
)
.
Lemma 4. The following equalities are true:
1.
µF̃
(
∆−(dn)
c1c2...cn
)
=
n∏
j=1
p̃cj ,j ≥ 0.
2. Let x0 = ∆−(dn)
ε1ε2...εn... be a nega-(dn)-irrational point, then
F̃
′
(x0) = lim
n→∞
n∏
j=1
dj p̃εj ,j
.
P r o o f.
1. Calculate the change µF̃ in the function F̃ on the cylinders ∆−(dn)
c1c2...cn . That
is, on the following closed intervals:
[
∆−(dn)
c1c2...c2n−1[d2n−1]0[d2n+2−1]0[d2n+4−1]...; ∆
−(dn)
c1c2...c2n−10[d2n+1−1]0[d2n+3−1]...
]
,
[
∆−(dn)
c1c2...c2n0[d2n+2−1]0[d2n+4−1]...;∆
−(dn)
c1c2...c2n[d2n+1−1]0[d2n+3−1]0[d2n+5−1]...
]
.
µF̃
(
∆−(dn)
c1c2...c2n−1
)
= F̃
(
∆−(dn)
c1c2...c2n−10[d2n+1−1]0[d2n+3−1]...
)
Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1 73
S.O. Serbenyuk
−F̃
(
∆−(dn)
c1c2...c2n−1[d2n−1]0[d2n+2−1]0[d2n+4−1]...
)
=
2n−1∏
j=1
p̃cj ,j
×(βd2n−1,2n+βd2n+1−1,2n+1pd2n−1,2n+βd2n+2−1,2n+2pd2n−1,2npd2n+1−1,2n+1+. . .)
=
2n−1∏
j=1
p̃cj ,j
(1− pd2n−1,2n + (1− pd2n+1−1,2n+1)pd2n−1,2n
+(1− pd2n+2−1,2n+2)pd2n−1,2npd2n+1−1,2n+1 + . . .) =
2n−1∏
j=1
p̃cj ,j
.
Analogously,
µF̃
(
∆−(dn)
c1c2...c2n
)
= F̃
(
∆−(dn)
c1c2...c2n[d2n+1−1]0[d2n+3−1]0[d2n+5−1]...
)
−F̃
(
∆−(dn)
c1c2...c2n0[d2n+2−1]0[d2n+4−1]...
)
=
2n∏
j=1
p̃cj ,j
(βd2n+1−1,2n+1 + βd2n+2−1,2n+2pd2n+1−1,2n+1
+βd2n+3−1,2n+3pd2n+1−1,2n+1pd2n+2−1,2n+2 + . . .) =
2n∏
j=1
p̃cj ,j
.
So,
µF̃
(
∆−(dn)
c1c2...cn
)
=
n∏
j=1
p̃cj ,j
≥ 0.
2. Find the derivative of F̃ at the nega-(dn)-irrational point x0 = ∆−(dn)
ε1ε2...εn....
Since
x0 = ∆−(dn)
ε1ε2...εn... =
∞⋂
n=1
∆−(dn)
ε1ε2...εn
,
we have
F̃
′
(x0) = lim
n→∞
µF̃
(
∆−(dn)
ε1ε2...εn
)
|∆−(dn)
ε1ε2...εn |
= lim
n→∞
∏n
j=1 p̃εj ,j
1
d1d2...dn
= lim
n→∞
n∏
j=1
dj p̃εj ,j
=
∞∏
j=1
(
dj p̃εj ,j
)
.
74 Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1
Continuous Functions with Complicated Local Structure
Since the function under consideration is continuous and monotonic (by the
Lebesgue theorem), it has a finite derivative almost everywhere in the sense of
the Lebesgue measure. But F̃
′
(x0) = ∞ in the case when the condition an =
dnp̃εn,n > 1 holds for all positive integers n except perhaps a finite number of n.
Therefore,
• if an ≥ 1 holds for a finite set of values n, then F̃
′
(x0) = 0;
• if an = 1 for all n ∈ N (it is true only for F̃ (x) = x), then F̃
′
(x0) = 1;
• if pεn,n 6= 1
dn
holds only for a finite set of values n, then 0 ≤ F̃
′
(x0) < ∞.
7. Nondifferentiable Functions
Let pi,n ∈ (−1; 1) for all n ∈ N, i = 0, dn − 1.
In this case, it follows from the statement 1 of Lemma 4 that the function
F̃ does not have any arbitrary small monotonicity interval if for each n ∈ N the
numbers pi,n, where i = 0, dn − 1, are either non-negative or negative.
Theorem 6. Let pεn,npεn−1,n < 0 for all n ∈ N, εn ∈ Adn \ {0}, 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 F̃ is nowhere differentiable on [0; 1].
P r o o f. Choose some nega-(dn)-rational point x0:
x0 = ∆−(dn)
ε1ε2...εn−1εn[dn+1−1]0[dn+3−1]... = ∆−(dn)
ε1ε2...εn−1[εn−1]0[dn+2−1]0[dn+4−1]...,
where εn 6= 0.
Let us introduce some notations. Let n be odd, then
x0 =x
(1)
0 =∆−(dn)
ε1ε2...εn−1εn[dn+1−1]0[dn+3−1]...=∆−(dn)
ε1ε2...εn−1[εn−1]0[dn+2−1]0[dn+4−1]...=x
(2)
0
and
x0 =x
(1)
0 =∆−(dn)
ε1ε2...εn−1[εn−1]0[dn+2−1]0[dn+4−1]...=∆−(dn)
ε1ε2...εn−1εn[dn+1−1]0[dn+3−1]...=x
(2)
0
in the case of even number n.
Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1 75
S.O. Serbenyuk
Let us consider the sequences (x
′
k), (x
′′
k) such that
x
′
k =
∆−(dn)
ε1...εn−1εn[dn+1−1]0[dn+3−1]0...[dn+k−1−1]1[dn+k+1−1]0[dn+k+3−1]...,
n ∈ O, k ∈ E;
∆−(dn)
ε1...εn−1εn[dn+1−1]0...[dn+k−2−1]0[dn+k−2]0[dn+k+2−1]0[dn+k+4−1]...,
n ∈ O, k ∈ O;
∆−(dn)
ε1...εn−1[εn−1]0[dn+2−1]0[dn+4−1]0...[dn+k−1−1]1[dn+k+1−1]0[dn+k+3−1]...,
n ∈ E, k ∈ O;
∆−(dn)
ε1...εn−1[εn−1]0[dn+2−1]0...[dn+k−2−1]0[dn+k−2]0[dn+k+2−1]0[dn+k+4−1]...,
n ∈ E, k ∈ E,
x
′′
k =
∆−(dn)
ε1...εn−1[εn−1]0[dn+2−1]0...[dn+k−1−1]00[dn+k+2−1]0[dn+k+4−1]...,
n ∈ O, k ∈ O;
∆−(dn)
ε1...εn−1[εn−1]0[dn+2−1]0...[dn+k−1][dn+k+1−1]0[dn+k+3−1]0[dn+k+5−1]...,
n ∈ O, k ∈ E;
∆−(dn)
ε1...εn−1εn[dn+1−1]0[dn+3−1]...0[dn+k−1][dn+k+1−1]0[dn+k+3−1]...,
n ∈ E, k ∈ O;
∆−(dn)
ε1...εn−1εn[dn+1−1]0...[dn+k−1−1]00[dn+k+2−1]0[dn+k+4−1]0[dn+k+6−1]...,
n ∈ E, k ∈ E,
where O is a set of all odd positive integers and E is a set of all even positive
integers.
That is,
x
′
k = x
(1)
0 +
1
d1d2 . . . dn+k
,
x
′′
k = x
(2)
0 − 1
d1d2 . . . dn+k
,
and x
′
k → x0, x
′′
k → x0 as k →∞.
Let n be an odd number. Then
y
(1)
0 = g(x(1)
0 ) = ∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]ε5...[dn−1−1−εn−1]εn(0),
y
(2)
0 =g(x(2)
0 )=∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]...[dn−1−1−εn−1][εn−1][dn+1−1][dn+2−1][dn+3−1]...,
y
′
k = g(x
′
k)=∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]...[dn−1−1−εn−1]εn 0...0︸︷︷︸
k−1
1(0)
,
y
′′
k =g(x
′′
k)=∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]...[dn−1−1−εn−1][εn−1][dn+1−1][dn+2−1]...[dn+k−1](0),
where F̃ (x) = F (g(x)) = F ◦ g.
76 Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1
Continuous Functions with Complicated Local Structure
Hence,
F̃ (x
′
k) = F (y
′
k) = βε1,1 +
n−1∑
t=2
β̃εt,t
t−1∏
j=1
p̃εj ,j
+ βεn,n
n−1∏
j=1
p̃εj ,j
+
(
n+k−1∑
l=n+1
(
β0,l
l−1∏
m=n+1
p0,m
))
n∏
j=1
p̃εj ,j
+ β1,n+k
n∏
j=1
p̃εj ,j
(
n+k−1∏
m=n+1
p0,m
)
,
F̃ (x(1)
0 ) = F (y(1)
0 ) = βε1,1 +
n−1∑
t=2
β̃εt,t
t−1∏
j=1
p̃εj ,j
+ βεn,n
n−1∏
j=1
p̃εj ,j .
Therefore,
F̃ (x
′
k)− F̃ (x(1)
0 ) = β1,n+k
n∏
j=1
p̃εj ,j
(
n+k−1∏
m=n+1
p0,m
)
=
n∏
j=1
p̃εj ,j
(
n+k∏
m=n+1
p0,m
)
.
In addition,
F̃ (x(2)
0 ) = F (y(2)
0 ) = βε1,1 +
n−1∑
t=2
β̃εt,t
t−1∏
j=1
p̃εj ,j
+ βεn−1,n
n−1∏
j=1
p̃εj ,j
+pεn−1,n
n−1∏
j=1
p̃εj ,j
( ∞∑
l=n+1
[
βdl−1,l
l−1∏
m=n+1
pdm−1,m
])
,
F̃ (x
′′
k) = F (y
′′
k ) = βε1,1 +
n−1∑
t=2
β̃εt,t
t−1∏
j=1
p̃εj ,j
+ βεn−1,n
n−1∏
j=1
p̃εj ,j
+pεn−1,n
n−1∏
j=1
p̃εj ,j
(
n+k∑
l=n+1
[
βdl−1,l
l−1∏
m=n+1
pdm−1,m
])
.
Hence,
F̃ (x(2)
0 )− F̃ (x
′′
k) = pεn−1,n
n−1∏
j=1
p̃εj ,j
(
n+k∏
m=n+1
pdm−1,m
)
.
Let n be an even number. In this case,
y
(1)
0 = g(x(1)
0 ) = ∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]...εn−1[dn−εn](0),
Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1 77
S.O. Serbenyuk
y
(2)
0 = g(x(2)
0 ) = ∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]...εn−1[dn−εn−1][dn+1−1][dn+2−1]...,
y
′
k = g(x
′
k) = ∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]...εn−1[dn−εn]0 . . . 0︸ ︷︷ ︸
k−1
1(0)
,
y
′′
k = g(x
′′
k) = ∆D
ε1[d2−1−ε2]ε3[d4−1−ε4]...εn−1[dn−1−εn][dn+1−1][dn+2−1]...[dn+k−1](0).
Thus,
F̃ (x
′
k) = F (y
′
k) = βε1,1 +
n−1∑
t=2
β̃εt,t
t−1∏
j=1
p̃εj ,j
+ βdn−εn,n
n−1∏
j=1
p̃εj ,j
+β1,n+k
n−1∏
j=1
p̃εj ,j
(
n+k−1∏
m=n+1
p0,m
)
pdn−εn,n,
F̃ (x(1)
0 ) = F (y(1)
0 ) = βε1,1 +
n−1∑
t=2
β̃εt,t
t−1∏
j=1
p̃εj ,j
+ βdn−εn,n
n−1∏
j=1
p̃εj ,j .
Therefore,
F̃ (x
′
k)− F̃ (x(1)
0 ) = β1,n+k
n−1∏
j=1
p̃εj ,j
(
n+k−1∏
m=n+1
p0,m
)
pdn−εn,n
=
n−1∏
j=1
p̃εj ,j
(
n+k∏
m=n+1
p0,m
)
pdn−εn,n.
In addition,
F̃ (x(2)
0 ) = F (y(2)
0 ) = βε1,1 +
n−1∑
t=2
β̃εt,t
t−1∏
j=1
p̃εj ,j
+ βdn−1−εn,n
n−1∏
j=1
p̃εj ,j
+
n∏
j=1
p̃εj ,j
( ∞∑
l=n+1
[
βdl−1,l
l−1∏
m=n+1
pdm−1,m
])
,
F̃ (x
′′
k) = F (y
′′
k ) = βε1,1 +
n−1∑
t=2
β̃εt,t
t−1∏
j=1
p̃εj ,j
+ βdn−1−εn,n
n−1∏
j=1
p̃εj ,j
+
n∏
j=1
p̃εj ,j
(
n+k∑
l=n+1
[
βdl−1,l
l−1∏
m=n+1
pdm−1,m
])
.
78 Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1
Continuous Functions with Complicated Local Structure
Hence,
F̃ (x(2)
0 )− F̃ (x
′′
k) = pdn−1−εn,n
n−1∏
j=1
p̃εj ,j
(
n+k∏
m=n+1
pdm−1,m
)
.
Thus,
B
′
k =
F̃ (x
′
k)− F̃ (x0)
x
′
k − x0
=
(dnpεn,n)
(∏n−1
j=1 dj p̃εj ,j
) (∏n+k
m=n+1 dmp0,m
)
,
n is an odd;
(dnpdn−εn,n)
(∏n−1
j=1 dj p̃εj ,j
)(∏n+k
m=n+1 dmp0,m
)
,
n is an even.
B
′′
k =
F̃ (x0)− F̃ (x
′′
k)
x0 − x
′′
k
=
(dnpεn−1,n)
(∏n−1
j=1 dj p̃εj ,j
)(∏n+k
m=n+1 dmpdm−1,m
)
,
n is an odd;
(dnpdn−1−εn,n)
(∏n−1
j=1 dj p̃εj ,j
)(∏n+k
m=n+1 dmpdm−1,m
)
,
n is an even.
Let us denote b0,k =
∏n+k
m=n+1 dmp0,m and bdk−1,k =
∏n+k
m=n+1 dmpdm−1,m.
Since
∏n−1
j=1 dj p̃εj ,j = const, pεn,npεn−1,n < 0, pdn−εn,npdn−1−εn,n < 0 and the
sequences (b0,k), (bdk−1,k) do not converge to 0 simultaneously (by the statement
of the theorem), we obtain the following cases:
1. If the inequalities dkp0,k > 1 and dkpdk−1,k > 1 hold for all k ∈ N except
perhaps a finite set of numbers k, then one of the sequences B
′
k, B
′′
k tends
to ∞, and another sequence tends to −∞;
2. If one of the products of dkp0,k, dkpdk−1,k is greater than 1, and another is
less than 1 for all k ∈ N except perhaps a finite set of numbers k, then one
of the sequences B
′
k, B
′′
k tends to ±∞, and another sequence tends to 0;
3. If one of the products of dkp0,k, dkpdk−1,k is greater than 1, and another is
equal to 1 for all k ∈ N except perhaps a finite set of numbers k, then one
of the sequences B
′
k, B
′′
k tends to ±∞, and another sequence is constant;
4. If one of the products of dkp0,k, dkpdk−1,k is less than 1, and another is
equal to 1 for all k ∈ N except perhaps a finite set of numbers k, then one
of the sequences B
′
k, B
′′
k tends to 0, and another sequence is constant;
5. If the products of dkp0,k, dkpdk−1,k are equal to 1 for all k ∈ N, then the
sequences B
′
k, B
′′
k are different constant sequences since the inequalities
pεn,n 6= pεn−1,n, pdn−εn,n 6= pdn−1−εn,n by the conditions pεk,k ∈ (−1; 1)
and βεk,k > 0 for εk > 0.
Journal of Mathematical Physics, Analysis, Geometry, 2017, vol. 13, No. 1 79
S.O. Serbenyuk
Since limk→∞B
′
k 6= limk→∞B
′′
k holds in all possible cases, it follows that the
function F̃ is nowhere differentiable on [0; 1].
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