BORNOVA / İZMİR July 2017
YAŞAR UNIVERSITY
GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
MASTER THESIS
COUNTEREXAMPLES IN ANALYSIS
ECEM KARAGÖZOĞLU
THESIS ADVISOR: ASST. PROF. AHMET YANTIR
APPLIED MATHEMATICS
v ABSTRACT
COUNTEREXAMPLES IN ANALYSIS
Karagözoğlu, Ecem Msc, Applied Mathematics Advisor: Asst. Prof. Ahmet YANTIR
July 2017
It is very important to make argument in mathematics and to understand it. Generally, we get help from the theorems proving another. However in some special cases we utilize counter examples. Counterexamples are used to indicate a predicate or a theory is wrong. We also say counterexamples are a kind of opponent proof. Anything that is meant to be described cannot be more illustrative than a good example. For this reason, we make use of counterexamples in mathematics. The aim of this thesis is to make some proofs with the help of counterexamples.
Key Words: counterexample, differentiation, functions and limit, sequences and series, proof of counterexamples
vii ÖZ
TÜRKÇE BAŞLIK
ANALIZDE TERS ÖRNEKLER Karagözoğlu, Ecem
Yüksek Lisans, Uygulamalı Matematik Danışman: Yrd.Doç. Dr. Ahmet YANTIR
Temmuz 2017
Matematikte ispat yapmak ve ispatı anlamak çok önemlidir. Genellikle, ispat yaparken teoremlerden yararlanırız. Fakat bazı özel durumlarda karşıt örnekler ya da örneklemeler kullanırız. Karşıt örnekler bir önermenin ya da bir teorinin yanlış olduğunu göstermek için kullanılır. Karşıt örnekler için bir tür karşıt ispatta diyebiliriz. Anlatılan ya da anlatılmak istenen herhangi bir şey iyi bir örnekten daha iyi olamaz. Bu sebeple, matematikte karşıt örnekler kullanarak yardım alırız. Bu tezin amacı, karşıt örnekler yardımıyla bazı ispatlar yapmaktır.
Anahtar Kelimeler: ters örnekler, türev, fonksiyon ve limit, diziler ve seriler, ters örnekler ispatı
ix
ACKNOWLEDGEMENTS
First of all, I would like to thank my supervisor Assistant Professor Ahmet Yantır for his guidance and patience during this study.
I would like to express my enduring love to my parents, who are always supportive, loving and caring to me in every possible way in my life.
Ecem Karagözoğlu İzmir, 2017
xiii TABLE OF CONTENTS ABSTRACT ... v ÖZ ... v ACKNOWLEDGEMENTS ... vii TEXT OF OATH ... xi
TABLE OF CONTENTS ... xiii
SYMBOLS AND ABBREVIATIONS ... xv
CHAPTER 1 INTRODUCTION ... 1
CHAPTER 2 PRELIMINARIES ... 2
2.1. Definitions and Important Theorems ... 3
CHAPTER 3 COUNTEREXAMPLES IN ANALYSIS ... 9
3.1. Functions and Limit ... 9
3.2.Differentiation………....13
3.3. Series and Sequences ... 18
CHAPTER 4 CONCLUSIONS ... 21
xv
SYMBOLS AND ABBREVIATIONS ABBREVIATIONS:
ܥሾܽǡ ܾሿ The set of continuous functions defined on ሾܽǡ ܾሿ ܷܥሾܽǡ ܾሿ The set of uniformly continuous defined on ሾܽǡ ܾሿ ܣܥሾܽǡ ܾሿ The set of absolutely continuous defined on ሾܽǡ ܾሿ ݉݁ݏܣ Measure of ܣ
ܤሺܫሻ The set of bounded functions defined on ܫ.
ܥଵሺܫሻ The set of continuously differentiable functions defined on ܫǤ ܤܸሾܽǡ ܾሿ The set of functions having bounded variation property on ሾܽǡ ܾሿǤ
SYMBOLS:
Թ Real numbers. Է Rational numbers. Գ Natural numbers.
1
CHAPTER 1 INTRODUCTION
Counterexamples play an important role in mathematics. They show that given mathematical statement (hypothesis, conjecture, proposition, etc.) is not correct.
They are efficient tool for mathematicians. If a mathematician seek for counterexample before trying to prove a conjecture or hypothesis, this process will provide to save time an effort for her/him. Here are list of examples for this advantage in the history of very famous prime numbers theory:
x In history of mathematics, listing the prime numbers and finding a formula for primes atracted many mathematicians. For a natural numbers of the form
ʹଶ
ͳ
were beleieved to be prime number for many years until a counterexample was found. For ݊ ൌ ͷ, the number
ʹଶఱ ͳ ൌ ͶʹͻͶͻʹͻ is a composite number and product of 641 and 6700417.
x Another famous conjecture about prime numbers is Goldbach or Goldbach-Euler conjecture and it is still waiting to be solved. The statement of the conjecture is very simple: Every even number greater than 2 is the sum of two prime numbers. For example, ͳʹ ൌ ͷ ǡ ʹͲ ൌ ͵ ͳǡ and so on. No counterexample have been found for this conjecture up to the number 4 x 1014 . It is a 1 million $ problem.
Another advantage for counterexamples is to understand the importance of the hypothesis of the theorems. For example for the existence and uniqueness of differentiation equation of the form
2
ݕሺݔሻ ൌ ݕ
if the conditions are satistified it is guarateed that there exist a unique solution for a spesific region. However it does not mean that the solution is not unique if one of the condition fails to hold.
The aim of this thesis is to provide some counterexamples with solutions in limits, continuity, differentiation, sequences and series in order to clarify these concepts.
In chapter 2, we will give brief information about the concepts mentioned above and we prepare the reader for the next chapter.
Chapter 3 gives the counterexamples with solutions.
Finally we conclude the results of the thesis by conclusion chapter.
CHAPTER 2 PRELIMINARIES
In this chapter, we will introduce some preliminary definitions and theorems that will be used throughout the thesis. Definitions and theorems and make explanations with examples. Each definition and theorem will help us in using and proving in future chapters. These basic informations can be found in many mathematics textbooks. We use the sources (Olmsted, 2003; J.Appell) for the informations. In this chapter we do not give all the details, only definitions and required theorems are stated. For more information please see (Olmsted, 2003)
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2.1. DEFINITIONS AND IMPORTANT THEOREMS 2.1.1. Limit of a Function
Let ݂ሺݔሻ be defined on an open interval about ݔ, except possibly at ݔ itself. We say that the limit of ݂ሺݔሻ as ݔ approaches ݔ is the number ܮ, and write
௫՜௫బ݂ሺݔሻ ൌ ܮ
if, for every number ߝ Ͳ, there exists a corresponding number ߜ Ͳ such that for all ݔ,
Ͳ ൏ ȁݔ െ ݔȁ ൏ ߜ ֜ ȁ݂ሺݔሻ െ ܮȁ ൏ ߝ Example
௫՜௫బݔ ൌ ݔ
Let ߝ Ͳ be given. We must find ߜ Ͳ such that for all ݔ satisfying Ͳ ൏
ȁݔ െ ݔȁ ൏ ߜǡ implies ȁݔ െ ݔȁ ൏ ߝ. The implication will hold if ߜ equals ߝor any smaller positive number. This proves that
௫՜௫బݔ ൌ ݔ.
2.1.2. The Sandwich Theorem
Sandwich theorem (or squeeze theorem) is used to find the limits of some complicated functions.
Let the function ݂ሺݔሻ be bounded by ݃ሺݔሻ and ݄ሺݔሻ ; i.e, ݃ሺݔሻ ݂ሺݔሻ ݄ሺݔሻ for all ݔ in some open interval containing ܿ, except possibly at ݔ ൌ ܿ itself. Also suppose that ௫՜݃ሺݔሻ ൌ ௫՜݄ሺݔሻ ൌ ܮǤ Then ௫՜݂ሺݔሻ ൌ ܮǤ Example Given that ͳ െ௫ସమ ݑሺݔሻ ͳ ௫ଶమ for all ݔ ് Ͳǡ The limit ௫՜ݑሺݔሻ ൌ ͳ, since ௫՜ͳ െ ௫మ ସ ൌ ௫՜ͳ ௫మ ଶ ൌ ͳǡ no matter how complicated ݑ is.
4
2.1.3. Upper and Lower Limit of a Sequence
Let the least term ݄ of a sequence be a term which is smaller than all but a finite number of the terms which are equal to݄. Then ݄ is called the lower limit of the sequence.
Let the greatest term ܪ of a sequence be a term which is greater than all but a finite number of the terms which are equal to ܪ. Then ܪ is called the upper limit of the sequence.
The upper and lower limit of a sequence of real numbers ሼݔሽ (called also lim superior and lim inferior) can be defined in several ways and are denoted, respectively as ՜λݏݑݔ ՜λ݂݅݊ ݔ . Example If ݔ ൌ ሺെͳሻǡ then ݔ ൌ െͳܽ݊݀ ݔ ൌ ͳǤ If ݔ ൌ ݊ ሺെͳሻǡ then ݔ ൌ Ͳܽ݊݀ ݔ ൌ λǤ 2.1.4. Continuity
x A function ݂ሺݔሻ is said to be continuous at ݔ ൌ ݔ if for given ߝ Ͳ there exists ߜ Ͳ such that ȁݔ െ ݔȁ ൏ Ͳ implies ȁ݂ሺݔሻ െ ݂ሺݔሻȁ ൏ ߝǤ
x A function is said to be continuous on an interval ܫ, if it is continuous at every point of the ܫ. The set of continuous functions onሾܽǡ ܾሿ is denoted by ܥሾܽǡ ܾሿ x A function is said to be continuous function if it is continuous at each point of
its domain.
A continuous function need not to be continuous on every interval.
For example,ݕ ൌ ͳ ݔൗ is not continuous , but it is continuous over its domain ሺെλǡ Ͳሻ ሺͲǡ λሻ.
5 2.1.5. The Intermediate Value Theorem
Let ܽǡ ܾ be real numbers with ܽ ൏ ܾ, and let ݂ be a continuous function from ሾܽǡ ܾሿ to Թ such that ݂ሺܽሻ ൏ Ͳ and ݂ሺܾሻ Ͳ. Then there is some number ܿ א ሺܽǡ ܾሻ such that ݂ሺܿሻ ൌ Ͳ.
Corollary
Let ݂be a continuous function from some interval ሾܽǡ ܾሿ to Թ , such that ݂ሺܽሻ and ݂ሺܾሻ have opposite signs. Then there is some number ܿ between and ܾ such that ݂ሺܿሻ ൌ Ͳ.
2.1.6. Absolutely Continuity
A function ݂ǣ ሾܽǡ ܾሿ ՜ Թ is absolutely continuous on ሾܽǡ ܾሿ if for every ߝ Ͳ there exists a ߜ Ͳ such that
ȁ݂ሺܾሻ െ ݂ሺܽሻȁ ே
ୀଵ
൏ ߝ
for any finite collection { [ܽ, ܾ] : 1 ݅ ܰ} of non-overlapping subintervals [ܽ, ܾ] of ሾܽǡ ܾሿ with
ȁܾെ ܽȁ ே
ୀଵ
൏ ߜǤ
The set of absolutely continuous functions on ሾܽǡ ܾሿ is denoted by ܣܥሾܽǡ ܾሿ. On a compact setܭ of Թ,
ܣܥሺܭሻ ك ܷܥሺܭሻ ൌ ܥሺܭሻ
where ܷܥሺܭሻ denotes the set of uniformly continuous functions on ܭ.
Example
݂ሺݔሻ ൌ ቊ Ͳǡ ݔ ൌ Ͳ
6 2.1.7. The Derivative
The derivative of the function ݂ሺݔሻ with respect to the variable ݔ is the function ݂ᇱ whose value at ݔis
݂ᇱሺݔሻ ൌ ՜
݂ሺݔ ݄ሻ െ ݂ሺݔሻ ݄
provided the limit exists.
Example
Let us use this definition to differentiate ݂ሺݔሻ ൌ௫ିଵ௫ Here we have ݂ሺݔሻ ൌ௫ିଵ௫ and ݂ሺݔ ݄ሻ ൌ ௫ା
ሺ௫ାሻିଵ, so
݂
ᇱሺݔሻ ൌ
՜ ሺ௫ାሻିሺ௫ሻ ൌ
ೣశ ೣశషభି ೣ ೣషభ ൌ
՜ ଵ Ǥ
ሺ௫ାሻሺ௫ିଵሻି௫ሺ௫ାିଵሻ ሺ௫ାିଵሻሺ௫ିଵሻൌ
՜ ଵ Ǥ
ି ሺ௫ାିଵሻሺ௫ିଵሻൌ
՜ ିଵ ሺ௫ାିଵሻሺ௫ିଵሻൌ
ିଵ ሺ௫ିଵሻమ. 2.1.8. Differentiable FunctionA differentiable function of one variable is a function whose derivative exists at each point of its domain.
If ݂ǣ ሺܽǡ ܾሻ ՜ ܴ is differentiable at ܿ א ሺܽǡ ܾሻ, then f is continuous at c. Example
݂ሺݔሻ ൌ ȁݔȁ continuous but not differentiable at ݔ ൌ Ͳ. 2.1.9. Periodic Function
A function ݂ሺݔሻ is said to be periodic (or, when emphasizing the presence of a single period instead of multiple periods, singly periodic) with period if
݂ሺݔሻ ൌ ݂ሺݔ ݊ሻ
For ݊ ൌ ͳǡʹǡ ǥ For example, the sine function ݔ, illustrated above, is periodic with least period (often simply called the “period”) ʹߨ (as well as with period െʹɎǡ ͶɎǡ Ɏǡ Ǥ).
7 2.1.10. Bounded Variation
(a) The function ߙǣ ሾܽǡ ܾሿ ՜ ܴ is said to be of bounded variation on ሾܽǡ ܾሿ if and only if there is a constant ܯ Ͳ such that
ȁߙሺݔሻ
ୀଵ
െ ߙሺݔିଵሻȁ ܯ for all partitions ܲ ൌ ሼݔǡ ݔଵǡ ǥ ǡ ݔሽ of ሾܽǡ ܾሿ.
(b) If ߙǣ ሾܽǡ ܾሿ ՜ ܴ is of bounded variation on ሾܽǡ ܾሿ, then the total variation of ߙ on ሾܽǡ ܾሿ is defined by
ܸఈሺܽǡ ܾሻ ൌ ሼσୀଵȁߙሺݔሻ െ ߙሺݔିଵሻȁȁǡ ܲ ൌ ሼݔǡ ݔଵǡ ǥ ǡ ݔሽሾܽǡ ܾሿሽ .
The set of functions obeying the rules of bounded variation is denoted by ܤܸሾܽǡ ܾሿǤ
Example
If ߙǣ ሾܽǡ ܾሿ ՜ ܴ is monotonically increasing, then for any partition ܲ ൌ ሼݔǡ ݔଵǡ ǥ ǡ ݔሽ of ሾܽǡ ܾሿ. ȁߙሺݔሻ െ ߙሺݔିଵሻȁ ୀଵ ൌ ሾߙሺݔሻ െ ߙሺݔିଵሻሿ ୀଵ ൌ ߙሺݔሻ െ ߙሺݔሻ ൌ ߙሺܾሻ െ ߙሺܽሻ Thus ߙ is of bounded variation and ܸሺܽǡ ܾሻ ൌ ߙሺܾሻ െ ߙሺܽሻ.
2.1.11. Luzin-n-Property
A function ݂ continuous on an interval ሾܽǡ ܾሿ.
For any set ܧ ؿ ሾܽǡ ܾሿ of measure mesE =0 , the image of this set, ݂ሺܧሻ , also has measure zero.
x A function ݂ ء constant on ሾܽǡ ܾሿ such that ݂ᇱሺݔሻ ൌ Ͳ almost-everywhere on ሾܽǡ ܾሿ does not have the Luzin-n-property.
x If ݂ does not have the Luzin-n-property, then on ሾܽǡ ܾሿ there is a perfect set P of measure zero such that ݉݁ݏ݂ሺܲሻ Ͳ.
x An absolutely continuous function has the Luzin-N-property.
x If ݂ has the Luzin-N-property and has bounded variation on ሾܽǡ ܾሿ then ݂ is absolutely continuous on ሾܽǡ ܾሿǤ
8
x If ݂ does not decrease on ሾܽǡ ܾሿ and ݂ᇱ is finite on ሾܽǡ ܾሿ, then ݂ has the Luzin-N-property.
x In order that ݂ሺܧሻ be measurable for every measurable set ܧ ؿ ሾܽǡ ܾሿ it is necessary and sufficient that ݂ have the Luzin-N-property on ሾܽǡ ܾሿ.
x A function ݂ that has the Luzin-N-property has a derivative ݂ᇱ on the set for which any non-empty portion of it has positive measure.
x For any perfect nowhere-dense set ܲ ؿ ሾܽǡ ܾሿ there is a function ݂ having the Luzin-N-property on ሾܽǡ ܾሿ and such that ݂ᇱdoes not exist at any point of P. The concept of Luzin’s N-property can be generalized to functions of several variables and functions of a more general nature, defined on measure spaces. 2.1.12. Nowhere Dense Set
A nowhere dense set in a topological space is a set whose closure has empty
interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere.
Example
ܵ ൌ ቄଵǣ ݊ א Գቅ is nowhere dense in Թ: although the points get arbitrarily close to 0, the closure of the set is ܵ ሼͲሽ, which has empty interior.
2.1.13. Hamel Basis
A basis for the real numbers Թ, considered as a vector space over the rational Է, i.e., a set of real numbers ሼܷఈሽ such that every real number ߚ has a unique representation of the form
ߚ ൌ ݎ
ୀଵ ܷఈ where ݎ is rational and ݊ depends on ߚ.
The axiom of choice is equivalent to the statement: "Every vector space has a vector space basis," and this is the only justification for the existence of a Hamel basis.
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CHAPTER 3
COUNTEREXAMPLES IN ANALYSIS
In this chapter, we will see lots of counterexamples and we try to resolve them step by step. Especially, we analyze counterexamples in functions and limit, differentiation, series and sequences.
3.1. Functions and Limit
In this section we consider some illustrative counterexamples about limit and functions. As we know, limit and functions are so important in mathematics (especially in calculus). The main concepts of mathematical analysis is based on functions and limit.
3.1.1. A nowhere continuous function whose absolute value is everywhere continuous.
݂ሺݔሻ ൌ ൜െͳ݂݅ݔ݅ݏ݅ݎݎܽݐ݈݅݊ܽ ͳ݂݅ݔ݅ݏݎܽݐ݈݅݊ܽ
Let ݔ א ܴ. Since the interval ȁݔ െ ݔȁ ൏ ߜ involves both rational and irrational functions ȁ݂ሺݔሻ െ ݂ሺݔሻȁ ൏ ߝ does not hold ݔ satisfying ȁݔ െ ݔȁ ൏ ߜ ; if we choose ߝ ൏ ʹ.
Note that ȁ݂ሺݔሻȁ ൌ ͳ which is constant; hence continuous everywhere. 3.1.2. A bounded function having no relative extrema on a compact domain.
Let the compact domain be the closed interval ሾͲǡͳሿ and for א ሾͲǡͳሿ , define ݂ሺݔሻ ؠ ቐ ሺെͳሻ݊ ݊ ͳ Ͳ݂݅ݔ݅ݏ݅ݎݎܽݐ݈݅݊ܽǤ݂݅ݔ݅ݏݎܽݐ݈݅݊ܽǡ ݔ ൌ ݉ ݊ Τ ݈݅݊ݓ݁ݏݐݐ݁ݎ݉ݏǡ ݊ Ͳ ȁ݂ሺݔሻȁ ൏ ͳ ฺ ݂݅ݏܾݑ݊݀݁݀Ǥ Choose compact domain: ሾͲǡͳሿ.
For n even and ሺݔݎܽݐ݈݅݊ܽሻ݂ሺݔሻ ൌ ାଵ ൏ ͳ For n is odd ሺݔݎܽݐ݈݅݊ܽሻ ݂ሺݔሻ ൌାଵି െͳ
10 No relative extreme value.
3.1.3. A nonconstant periodic function without a smallest positive period. ݂ሺݔሻ ൌ ൜െͳ݂݅ݔ݅ݏ݅ݎݎܽݐ݈݅݊ܽ ͳ݂݅ݔ݅ݏݎܽݐ݈݅݊ܽ
The periods of any real-valued function with domain R form an additive group (that is, the set of periods is closed with respect to subtraction).
x ݂ is not constant
x ݂ is periodic since the graph of ݂ repeats itself for every interval.
Smallest possible periodؠ the difference between two consequtive rationals which does not exist.
3.1.4. A function continuous and one-to-one on an interval and whose inverse is not continuous.
Our example in this case is a complex-valued function ݖ ൌ ݂ሺݔሻ of the real variable ݔ, with continuity defined exactly as in the case of a real-valued function of a real variable, where the absolute value of the complex number ݖ ൌ ሺܽǡ ܾሻ is defined
ȁݖȁ ൌ ȁሺܽǡ ܾሻȁ ؠ ሺܽଶ ܾଶሻଵൗଶ Let the function ݖ ൌ ݂ሺݔሻ be defined:
ݖ ൌ ݂ሺݔሻ ؠ ሺܿݏݔǡ ݏ݅݊ݔሻǡ Ͳ أ ݔ ൏ ʹߨ.
݂ǣ ܴ ื ܥ ݔ ݖ ൌ ݂ሺݔሻ א ܥ ݂ሺݔሻ ൌ ܿݏݔ ݅ݏ݅݊ݔ
ȁݖȁ ൌ ͳܿ݉ܽܿݐǤ A continuous function converts a compact range into a compact range.
ሺͲǡʹߨሻ݊ݐܿ݉ܽܿݐ ฺ ݂ିଵ݅ݏ݊ݐܿ݊ݐ݅݊ݑݑݏ.
3.1.5. A function continuous at every irrational point and discontinuous at every rational point.
Let ݂ሺݔሻ be defined as follows:
If ݔ is a rational number equal to ݉ ݊Τ , where ݉ and ݊ are integers such that the fraction ݉ ݊Τ is in lowest terms and ݊ Ͳ, let ݂ሺݔሻ be defined to be equal to ͳ ݊Τ ; otherwise, if ݔ is irrational, let ݂ሺݔሻ ؠ Ͳǡ i.e
11 ݂ሺݔሻ ൌ ൜ͳ ݊ǡͲǡΤ ݔ݅ݎݎܽݐ݈݅݊ܽݔݎܽݐ݈݅݊ܽ Let ݔ א Է ȁݔ െ ݔȁ ൏ ߜ ֜ ฬ݂ሺݔሻ െ ͳ ݊ฬ ൏ ߝ ቚݔ െ݉ ݊ቚ ൏ ߜ ฺ ฬ݂ሺݔሻ െ ͳ ݊ฬ ൏ ߝ There is always an irrational on interval ݔെ ߜ ൏ ݔ ൏ ݔ ߜ
֜ ฬͳ ݊ฬ ൏ ߝ ֜ ݊ ൏ ͳ ߝ ֜ ݉ ݊ ൏ ߝ 3.1.6. A discontinuous linear function.
A function ݂ on ܴ into ܴ is said to be linear if and only if ݂ሺܽݔ ܾݕሻ ൌ ݂ܽሺݔሻ ܾ݂ሺݕሻ for all ݔǡ ݕǡ ܽǡ ܾ א ܴ. A function that is linear and not continuous is very complicated indeed.
Construction of a discontinuous linear function can be achieved by use of a Hamel basis for the linear space of the real numbers ܴ over the rational numbers Է. The idea is that this process provides a set ܵ ൌ ሼݎఈሽ of real numbers ݎఈ such that every real number ݔ is a unique linear combination of a finite number of members of ܵ with rational coefficients ఈǣ ݔ ൌ ଵݎଵ ڮ ݎ. The function ݂ can now be defined: ݂ሺݔሻ ؠ ଵ ڮ ݔ ൌ ͳǤ ሺݔ െ ͳሻ ʹǤʹ ͵Ǥ ሺെͳሻ ݂ሺݔሻ ൌ ͳ ʹ ͵ ൌ f is linear ݂ሺݔ ݕሻ ൌ ݂ሺ ఈݎఈ ݍ௫ݎఉሻ ൌ ݂ሺ ఈݎఈ ݍ௫ݎ௫ሻ ൌ ݂ሺሺఈ ݍ௫ሻ ݎ௫ሻ ൌ ሺఈ ݍఈሻ ൌ ఈ ݍఈ ൌ ݂ሺݔሻ ݂ሺݕሻ
Since ݂ሺݔሻ א Է, then it does not attain any irrational number between ݍଵ and ݍଶ. ֜ ݂ does not satisfy intermediate value property.
12
3.1.7. Functions ࢟ ൌ ࢌሺ࢛ሻǡ ࢛ א ࡾǡ ࢇࢊ࢛ ൌ ࢍሺ࢞ሻǡ ࢞ א ࡾ, whose composite function ࢟ ൌ ࢌሺࢍሺ࢞ሻሻ is everywhere continuous, and such that
ܔܑܕ ࢛՜࢈ࢌሺ࢛ሻ ൌ ࢉǡ ܔܑܕ࢞՜ࢇࢍሺ࢞ሻ ൌ ࢈ǡ ࢌ൫ࢍሺ࢞ሻ൯ ് ࢉǤ If ݂ሺݑሻ ൌ ቄͲǡݑ ് Ͳͳǡݑ ൌ Ͳ ݃ሺݔሻ ؠ Ͳ for all ݔ א ܴǤ ௫՜݃ሺݔሻ ൌ Ͳǡ ௨՜݂ሺݑሻ ൌ Ͳ But ௫՜݂൫݃ሺݔሻ൯ ൌ ௫՜݂ሺͲሻ ൌ ௫՜ͳ ൌ ͳǤ
This counterexample becomes impossible in case the following condition is added: ݔ ് ܽ ֜ ݃ሺݔሻ ് ܾǤ
3.1.8. Let ࡵ ൌ ሾǡ ሿ and ࢻǡ ࢼ א Թ, and let ࢌࢻǡࢼǣ ࡵ ՜ Թ be defined by ࢌࢻǡࢼሺ࢞ሻ ؔ ൜࢞ࢻܛܑܖ ࢞ࢌ࢘࢞ ൌ ࢼࢌ࢘ ൏ ࢞
ࢌࢻǡࢼ א ሺࡵሻ holds precisely for ࢻ and arbitrary ࢼǡ ࢘ࢻ ࢇࢊࢼ െࢻǤ x ߙ Ͳǡ ߚܽݎܾ݅ݐݎܽݎݕ ௫՜݂ఈǡఉ ൌ Ͳ Let ߙ Ͳǡ ߚ א ԹǤ െͳ ݔఉ ͳ ֜ െݔఈ ݔఈ ݔఉ ݔఈ Since ௫՜ݔ ఈ ൌ ͲǢ by sandwich theorem; ௫՜ݔ ఈǤ ݔఉ ൌ Ͳ ݂ఈǡఉሺݔሻ is continuous at ݔ ൌ ͲǤ ֜ ݂ఈǡఉ א ܥሺܫሻ . x ߙ ͲǢ ߚ െߙ Is ݂ఈǡఉ continuous at 0 ? ( ௫՜݂ఈǡఉሺݔሻ ൌ Ͳ) for ݔ ് Ͳ ݂ఈǡఉሺݔሻ ൌ ݔఈǤ ݔఉ , ߚ െߙ ֜ ߚ ൌ ȁߙȁ ߝǡ ߝ Ͳ
13
ൌ
ୱ୧୬ ௫௫ȁഀȁഁൌ
ୱ୧୬ ௫௫ȁഀȁȁഀȁశഄǤ௫ഄǤ௫ഄൌ ݔ
ఌǤ
ୱ୧୬ ௫ഁ ௫ഁ ௫՜݂ఈǡఉሺݔሻ ൌ ௫՜ݔ ఌǤୱ୧୬ ௫ഁ ௫ഁ ൌ ௫՜ݔ ఌ ᇣᇤᇥ ୀ Ǥ ௫՜ ௫ഁ ௫ഁ ᇣᇤᇥ ୀଵ ൌ ͲǤ ֜ ݂ఈǡఉ is continuous at ݔ ൌ ͲǢ֜ ݂ א ܥሺܫሻǤ 3.2. DIFFERENTIATION
We present some counterexamples about differentiation in this section. Differentiation in this part. differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.
3.2.1. A differentiable function with a discontinuous derivative. The function
݂ሺݔሻ ؠ ൝ ݔଶݏ݅݊ ͳ ݔ
Ͳ݂݅ݔ ൌ Ͳ݂݅ݔ ് Ͳǡ has as its derivative the function
݂ᇱሺݔሻ ൌ ቊʹݔ ͳ ݔൗ െ ͳ ݔൗ ݂݅ݔ ് Ͳ Ͳ݂݅ݔ ൌ Ͳ ௫՜݂ ᇱሺݔሻ ൌǫ െͳ ൏ ͳ ݔ൏ ͳ ֜ െʹݔ ൏ ʹݔ ͳ ݔ൏ ʹݔ ֜ ௫՜ᇣᇧᇤᇧᇥെʹݔ ୀ ൏ ௫՜ǥ ൏ ᇣᇧᇤᇧᇥ ୀ ௫՜ᇣᇤᇥʹݔ ୀ െͳ ൏ ௫՜ ଵ ௫൏ ͳǡ which means ֜ ௫՜݂
ᇱሺݔሻ does not exist. Therefore, ݂ᇱሺݔሻ is discontinuous at ݔ ൌ ͲǤ
3.2.2. A differentiable function for which the law of the mean fails. The complex-valued function of a real variable ݔ,
14
is everywhere continuous and differentiable but there exist no ܽǡ ܾǡ ܽ݊݀ߦ such that ܽ ൏ ߦ ൏ ܾ and ሺ ܾ ݅ ܾሻ െ ሺ ܽ ݅ ܽሻ ൌ ሺെ ߦ ݅ ߦሻ ሺܾ െ ܽሻǤ By the law of mean, ݂ must be continuous on ሾܽǡ ܾሿ and differentiable on ሺܽǡ ܾሻǤ Then, ܿሺܽǡ ܾሻ such that
݂ሺܾሻ െ ݂ሺܽሻ
ܾ െ ܽ ൌ ݂ᇱሺܿሻ ݂ሺݔሻ ൌ ݔ ݅ ݔ , continuous and differentiable
݂ᇱሺݔሻ ൌ െ ݔ ݅ ݔ ܾ ݅ ܾ െ ܽ ݅ ܽ
ܾ െ ܽ ൌ െ ܿ ݅ ܿǤሺͳሻ Assume that (1) is correct.
ܾ ݅ ܾ െ ܽ െ ݅ ܽ ൌ ሺܾ െ ܽሻǤ ሺെ ܿ ݅ ܿሻ ֜ ሺ ܾ െ ܽሻ ݅ሺ ܾ െ ܽሻ ൌ ሺܾ െ ܽሻሺെ ܿ ݅ ܿሻ ሺ ܾ െ ܽሻଶ ሺ ܾ െ ܽሻଶ ൌ ሺܾ െ ܽሻଶ
֜
ሺିሻమ ସൌ
ୡ୭ୱమାୡ୭ୱమିଶ௦௦ାୱ୧୬మାୱ୧୬మିଶ௦௦ ସൌ ʹሺͳ െ ሺ ܽ ܾ ܽ ܾሻሻ Ͷ ൌ ʹǤͳ െ ሺܾ െ ܽሻ Ͷ ൌ ͳ െ ሺܾ െ ܽሻ ʹ ൌ ͳ െ ሺͳ െ ʹݏ݅݊ଶܾ െ ܽ ʹ ሻ ʹ ൌ ݏ݅݊ଶ ሺܾ െ ܽሻ ʹ Ǥ ሺܾ െ ܽሻ Ͷ ଶ ൌ ݏ݅݊ଶ൬ܾ െ ܽ ʹ ൰ ֜ ൬ ܾ െ ܽ ʹ ൰ ଶ ൌ ݏ݅݊ଶሺܾ െ ܽ ʹ ሻ ֜ ܾ െ ܽ ʹ ൌ ܾ െ ܽ ʹ From this equation, ݔ ൌ ݔ but this is a contradiction.
Hence, no solution for ݔ ͲǤ
3.2.3. Let ࡵ ൌ ሾǡ ሿ and ࢻǡ ࢼ א Թ, and let ࢌࢻǡࢼǣ ࡵ ՜ Թ be defined by ࢌࢻǡࢼሺ࢞ሻ ؔ ൜࢞ࢻܛܑܖ ࢞ࢌ࢘࢞ ൌ ࢼࢌ࢘ ൏ ࢞
ࢌࢻǡࢼ א ࡰሺࡵሻ holds precisely for ࢻ and arbitrary ࢼǡ ࢘ࢻ ࢇࢊࢼ െ ࢻǤ
x ͳܽ݊݀ߚarbitrary
For ് Ͳ , ݂ఈǤఉሺݔሻ א ܦଵሺܫሻ this statement is trivial.
݂ఈǡఉᇱ ሺͲሻ ൌ ՜శ ݂ఈǡఉሺͲ ݄ሻ െ ݂ᇩᇪᇫఈǡఉሺͲ ୀ ሻ ݄
15 ൌ ՜ ݂ఈǡఉሺ݄ሻ ݄ ൌ ՜ ݄ఈǤ ݄ఉ ݄ ൌ ՜శ݄ ఈିଵǤ ݄ఉ ߙ ͳ ֜ ߙ െ ͳ Ͳ ֜ ݄ఈିଵ Ͳ ͳ ݄ఉ ͳǡ ߚ ֜ ݄ఈିଵ ݄ఈିଵǤ ݄ఉ ݄ఈିଵ ՜݄ ఈିଵൌ Ͳ ֜ ܤݕݏܽ݊݀ݓ݄݅ܿݐ݄݁ݎ݁݉ ݂ఈǡఉᇱ ൌ ՜శ݄ ఈିଵ ݄ఉ ൌ Ͳ ֜ ݂ఈǡఉ א ܦଵሺܫሻǤ x ߙ ͳܽ݊݀ߚ ͳ െ ߙǤ ݂ᇱሺͲሻ ൌ ڮ ൌ ՜݄ ఈିଵǤ ݄ఉ ߙ ͳ ֜ ߙ െ ͳ Ͳ ͳ െ ߙ Ͳ ߚ ͳ െ ߙ ֜ ߚ ൌ ͳ െ ߙ ߝǢ ߝ Ͳ ݂ᇱሺͲሻ ൌ ڮ ൌ ՜ ୱ୧୬ ഁ భషഀ ൌ ՜ ୱ୧୬ ഁǤഄ భషഀǤഄ ൌ ՜శ ୱ୧୬ ഁǤഄ ഁ ൌ ՜శ݄ ఌ ᇣᇧᇤᇧᇥ ୀ Ǥ ՜శ ୱ୧୬ ഁ ഁ ᇣᇧᇧᇤᇧᇧᇥ ୀଵ ൌ Ͳ ֜ ݂ᇱሺͲሻ exists ֜ ݂ א ܦଵሺܫሻ exists.
16
3.2.4. Let ࡵ ൌ ሾǡ ሿ and ࢻǡ ࢼ א Թ, and let ࢌࢻǡࢼǣ ࡵ ՜ Թ be defined by ࢌࢻǡࢼሺ࢞ሻ ؔ ൜࢞ࢻܛܑܖ ࢞ࢼࢌ࢘ ൏ ࢞
ࢌ࢘࢞ ൌ ࢌࢻǡࢼ א ሺࡵሻ holds precisely for arbitrary ࢻ and ࢼ െ ࢻǤ a) Let ݔ א ሺͲǡͳሿ ֜ ݂ఈǡఉᇱ ሺݔሻ ൌ ߙǤ ݔఈିଵǤ ݔఉ ݔఈǤ ݔఉǤ ߚǤ ݔఉିଵ ֜ ݂ఈǡఉᇱ ሺݔሻ ൌ ߙǤ ݔఈିଵǤ ݔఉ ߚǤ ݔఉିଵǤ ݔఈǤ ߚ ݂ఈǡఉᇱ ሺݔሻ ൌ ߙǤ ݔఈିଵǤ ݔఉ ߚǤ ݔఈାఉିଵǤ ݔఉ x ߙ െ ͳ Ͳ ֜ Ͳ ൏ ݔఈିଵ ͳ and െͳ ൏ ݔఉ ͳ ֜ ߙǤ ݔఈିଵǤ ߚ א ܤሺሺͲǡͳሿሻ ߙ െ ͳ ᇣᇤᇥ ֜ଵିఈழ Ͳ ֜ ߚ ߙ െ ͳ ͳ െ ߙ ݔఈାఉିଵ ݔଵିఈ א ܤሺͲǡͳሻ ֜ ݔఈାఉିଵ א ܤሺܫሻ ֜ ߚǤ ݔఈାఉିଵǤ ݔఉ א ܤሺሺͲǡͳሿሻ ֜ ݂ఈǡఉᇱ ሺݔሻ א ܤ൫ሺͲǡͳሿ൯݂ݎߙ െ ͳ ͲǤሺͳሻ x ߙ െ ͳ ൏ Ͳ ֜ ߙ ൏ ͳ ֜ Ⱦ ͲǤ ݂ఈǡఉᇱ ሺݔሻ ൌ ߙǤ ݔถఈିଵ ୀଵ ௫Τ భషഀ Ǥ ݔᇣᇤᇥఉ אሾିଵǡଵሿ ߚǤ ݔᇣᇧᇤᇧᇥఈାఉିଵ אሺǡଵሿ Ǥ ݔᇣᇤᇥఉ אሾିଵǡଵሿ ߙ ߚ െ ͳ ߙ െ ͳ ͳ െ ߙ ൌ Ͳ ֜ ݂ఈǡఉᇱ ሺݔሻ א ܤ൫ሺͲǡͳሿ൯݂ݎߙ െ ͳ ൏ Ͳሺʹሻ ֜ ݂ఈǡఉᇱ ሺݔሻ א ܤ൫ሺͲǡͳሿ൯ߙܽ݊݀ߚ ͳ െ ߙ b) Let ݔ ൌ ͲǤ ݂ఈǡఉᇱ ሺͲሻ ൌ ՜ ഀୱ୧୬ ഁି ൌ ՜ ݄ఉ ݄ଵିఈ ൌ ՜ ݄ఉ ݄ఉିఌ ൌ ՜ ݄ఉǤ ݄ఌ ݄ఉ ൌ Ͳ By (a) and (b) ݂ఈǡఉᇱ ሺݔሻ א ܤሺሾͲǡͳሿሻߙܽ݊݀ߚ ͳ െ ߙ ݂ఈǡఉ א ܤଵሺሾͲǡͳሿሻ
17
3.2.5. Let ࡵ ൌ ሾǡ ሿ and ࢻǡ ࢼ א Թ, and let ࢌࢻǡࢼǣ ࡵ ՜ Թ be defined by ࢌࢻǡࢼሺ࢞ሻ ؔ ൜࢞ࢻܛܑܖ ࢞ࢼࢌ࢘ ൏ ࢞
ࢌ࢘࢞ ൌ
ࢌࢻǡࢼ א ሺࡵሻ holds precisely for arbitrary ࢻ and ࢼ െ ࢻǤ x ݔ א ሺͲǡͳሿǤ ݂ఈǡఉᇱ ൌ ߙǤ ݔఈିଵǤ ݔఉ ߚǤ ݔఈାఉିଵǤ ݔఉ א ߝሺሺͲǡͳሿሻ ߚ ͳ െ ߙ ֜ ߙ ߚ െ ͳ ͳ െ ߙ ߙ െ ͳ ൌ ͲǤ x ݂ఈǡఉᇱ ሺͲሻ ൌ Ͳ. ௫՜ ݂ఈǡఉ ᇱ ሺݔሻ ൌ ݂ ఈǡఉᇱ ሺͲሻ correct ? ௫՜ߙǤ ݔ ఈିଵǤ ݔఉ ߚǤ ݔఈାఉିଵǤ ݔఉ ൌ ௫՜ߙǤ ݔఉ ݔଵି௫ ߚ ᇣᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇥ௫՜ݔఈାఉିଵǤ ݔఉ ୀ ൌ ͲǤ ݂ఈǡఉᇱ ሺݔሻ is continuous ݔ א ሾͲǡͳሿǤ ֜ ݂ఈǡఉሺݔሻ א ܥଵሺሾͲǡͳሿሻǤ
18 3.3. SERIES AND SEQUENCES
3.3.1. A divergent series whose general term approaches zero. Let us choose the harmonic series.
ͳ ݊ ஶ ୀଵ Clearly ՜ஶ ଵ ൌ Ͳ.
Now we will show that σஶୀଵଵ is divergent. There are several ways to show the divergence of this series such as integral test, and geometric way etc. However we will show this analitically.
Suppose that ͳ ݊ ஶ ୀଵ ൌ ܮ i.e, ܮ ൌ ͳ ͳ ʹ ͳ ͵ ͳ Ͷ ڮ Clearly ܮ ൌ ͳ ͳ ʹ ͳ ͵ ͳ Ͷ ͳ ͷ ڮ ͳ ͳ ʹ ͳ Ͷ ͳ Ͷ ͳ ͳ ͳ ͺ ͳ ͺ ڮ ൌ ͳ ͳ ʹ ͳ ʹ ͳ ͵ ͳ Ͷ ͳ ͷ ڮ ൌ ܮ ͳ ʹ
֜ ܮ ܮ ଵଶ which is a contradiction. Hence the series does not converge.
3.3.2. Bounded Divergent Sequences.
The simplest example of a bounded divergent sequence is possibly ͲǡͳǡͲǡͳǡ ǥ
or ሼܽሽ, where ܽ ൌ Ͳ if ݊ is odd and ܽ ൌ ͳ is ݊ is even. Equivalently, ܽ ൌ ଵ
ଶሺͳ ሺെͳሻሻǤ
Let ሼܽሽ be the sequence in ܴ defined as ܽ ൌ ሺെͳሻ. It is clear that ሼܽሽ is bounded above by ͳ and below by െͳ. Note the following subsequences of ሼܽሽ
19
x ሼܽሽ where ݊ is the sequence defined as ݊ ൌ ʹݎ x ሼܽሽ where ݊௦ is the sequence defined as ݊௦ ൌ ʹݏ ͳ The first is ͳǡͳǡͳǡ ǥand the second is െͳǡ െͳǡ െͳǡ ǥ
So, ሼܽሽ has two subsequences with different limits. Since any two subsequence of a sequence cannot converge to different limits, ሼܽሽ cannot be convergent.
3.3.3. A divergent sequence ሼࢇሽ for which ܔܑܕ
՜ĞĞሺ ࢇାെ ࢇሻ ൌ for every
positive integer Ǥ
Let ܽ be the ݊ݐ݄ partial sum of the harmonic series. ܽ ؠ ͳ
ͳ
ʹ ڮ ͳ ݊Ǥ Then ሼܽሽ is divergent, but for Ͳǡ
ܽାെ ܽ ൌ ͳ ݊ ͳ ڮ ͳ ݊ ݊ ͳ՜ ͲǤ ՜ஶሺܽାെ ܽሻ ൌ ͲǤ
3.3.4. Sequences ሼࢇሽ and ሼ࢈ሽ such that ܔܑܕܑܖሺࢇሻ ܔܑܕܑܖሺ࢈ሻ ൏ ܔܑܕܑܖሺࢇ ࢈ሻ ൏ ܔܑܕܑܖሺࢇሻ ܔܑܕܛܝܘሺ࢈ሻ ൏ ܔܑܕܛܝܘሺࢇ ࢈ሻ ൏ ܔܑܕܛܝܘሺࢇሻ ܔܑܕܛܝܘሺ࢈ሻǤ
Let ሼܽሽ and ሼܾሽ be the sequences repeating in cycles of 4:
ሼܽሽ Ͳǡ ͳǡ ʹǡ ͳǡ Ͳǡ ͳǡ ʹǡ ͳǡ ǥ ฺ ሺܽሻ ൌ Ͳǡ ݏݑ ሺܽሻ ൌ ʹ ሼܾሽ ʹǡ ͳǡ ͳǡ Ͳǡ ʹǡ ͳǡ ͳǡ Ͳǡ ǥ ฺ ሺܾሻ ൌ Ͳǡ ሺܾሻ ൌ ʹ Now, we sum up the expression
ܽ ܾ ൌ ʹǡ ʹǡ ͵ǡ ͳǡ ʹǡ ʹǡ ͵ǡ ͳǡ ǥ ሺܽ ܾሻ ൌ ͳ
ሺܽ ܾሻ ൌ ͵ Hence,
21 CHAPTER 4 CONCLUSIONS
This thesis tries to emphasize the importance of counterexamples in mathematics. As in stated in Introduction chapter, counterexamples are very important tool to show that a given mathematical statement is incorrect.
The subjects are choosen from calculus in order to be more understandable. These examples also emphasisez the importance of premises of the theorems.
We aim to give very basic and illustrative examples to be clear. People who are interested in counterexamples in analysis may read the reference (Olmsted,2003) which is involves huge amount of examples in every branch of mathematical analysis.
In other branches of mathematics, such as algebra, topology etc., there are many sources about counterexamples.
22
REFERENCES
Gelbaum, B. R., & Olmsted J. M. H. (1992). Counterexamples in Analysis. NewYork, Dover.
Appell J. Some Counterexamples for Your Calculus Course, Würzburg. Thomas, G. B. (2005). Thomas’ Calculus. Addison-Wesley.
Counterexample. Retrieved December 2016, from
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