Book by Jaromir Kuben

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c 2012, Jaromr Kuben
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Preface

This textbook is written for undergraduate students studying mathematics in English, at the Faculty ofMilitary Technology, University of Defence. It contains an explanation of differential calculus of func-tions of a single variable, which, together with integral calculus, underlie math education in the Ing.degree programme. Knowledge of differential calculus is a prerequisite for other areas of mathematicssuch as integral calculus of functions of a single variable, differential equations, differential and integralcalculus of functions of several variables, and vector analysis. It is one of the critical building blocks ofmechanics, physics, and other applied disciplines.

Looking into contemporary textbooks of differential andintegral calculus of functions of one variablewe find that they usually begin with descriptions of real numbers. This is followed by limits, and the useof limits to define the derivative. Only then are the indefinite and finally the definite integral explored.

However, these concepts did not arise in this order historically. The concept of the definite integral(evaluating areas and volumes) was developed first, followed in the 17th century by the derivative andindefinite integral, which were based on intuitive understandings of infinitely small and big quantities.It was not until the following century that the concept of limit was elaborated, and another century after

that, that the theory of real numbers was posited.3


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for all exercises. To improve orientation within the text, the ends of proofs are denoted by the symboland ends of examples by the symbol .

There is a long list of textbooks devoted to differential calculus of functions of a single variable. Allproofs missing in this text can be found in [1], unless a different source is mentioned. [3] is a classicalCzech monograph. [11,16,17] are worth a look, and for Russian speakers the classical textbook [2] canbe consulted.

I thank the reviewers, Prof. RNDr. Zuzana Dol, DSc. and Doc. RNDr. Jaromr ima, CSc.of the Department of Mathematics and Statistics of the Faculty of Science, Masaryk University, Brnofor their invaluable comments and contribution to the quality of the text. I am especially grateful to

my colleague PhDr. Pavlna Rakov, Ph.D. Not only did she carefully review the entire text, correctnumerous mistakes, and prepare four quizzes, she also shouldered many of my duties over the last half--year, allowing me to complete this text in a timely manner. I am much obliged for all her support. I alsothank my colleague Ing. Ji Jnsk, Ph.D., who contributed some exercises and my son Bc. JaromrKuben, who prepared nine quizzes. Finally, I thank Robert Brukner, MA, for his thorough copy edit andlanguage correction of the text.

The text was prepared with the pdfTEX typesetting system using the LATEX 2 format. The figureswere created with METAPOSTusing the T

EX macro package mfpic.

The creation of this textbook was supported by ESF under the Innovation of the Military TechnologyStudy Programme(reg.num. CZ.1.07/2.2.00/07.0256) of the University of Defence, Brno. Aside fromthis electronic, fully hypertexted version, hard copy [6]also exists. It precedes the textbook [7] and itshypertexted version, which were also supported by ESF.

Brno, January 2012 Jaromr Kuben5


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Contents

Preface 3

1 Basic Concepts 13

1.1 Structure of Mathematical Texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 Number sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.1 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.2 Properties of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.3 Powers and Roots of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 25

1.4 Extended Set of Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.5 Mappings and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34QuizBasic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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2 Real Functions of a Single Variable 37

2.1 Definition of a Function and its Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1.1 Definition of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.1.2 Equality of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.1.3 Graph of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2 Properties of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.2.1 Bounded Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.2.2 Monotonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.2.3 One-to-one Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.2.4 Even and Odd Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.2.5 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3 Operations with Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.3.1 Sums, Differences, Products, and Quotients of Functions . . . . . . . . . . . . 592.3.2 Composition of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.3.3 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.4 Transforming Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77QuizReal functions of a single variable . . . . . . . . . . . . . . . . . . . . . . . . 78

3 Elementary Functions 81

3.1 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 823.1.1 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.1.2 Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957


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QuizExponential, logarithmic, and power functions . . . . . . . . . . . . . . . . . . 973.2 Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.2.1 General Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.2.2 Special Cases of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.3 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.3.1 Definition of Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . 1123.3.2 Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143.3.3 Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.3.4 Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.3.5 Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.3.6 Secant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.3.7 Cosecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.4 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.4.1 Inverse Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.4.2 Inverse Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.4.3 Inverse Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.4.4 Inverse Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303.4.5 Inverse Secant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.4.6 Inverse Cosecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

3.5 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583.5.1 Hyperbolic Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

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3.5.2 Hyperbolic Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593.5.3 Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

3.5.4 Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.5.5 Hyperbolic Secant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623.5.6 Hyperbolic Cosecant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

3.6 Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.6.1 Inverse Hyperbolic Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.6.2 Inverse Hyperbolic Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1663.6.3 Inverse Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . 1673.6.4 Inverse Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . 168

3.6.5 Inverse Hyperbolic Secant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1693.6.6 Inverse Hyperbolic Cosecant. . . . . . . . . . . . . . . . . . . . . . . . . . . 170Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172QuizTrigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic func-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1733.7 Polynomials and Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

3.7.1 Basic Concepts and Operations with Polynomials . . . . . . . . . . . . . . . . 175

3.7.2 Horners Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1793.7.3 Roots of Polynomials and Factorization . . . . . . . . . . . . . . . . . . . . . 1823.7.4 Finding of Roots of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . 1873.7.5 Integer and Rational Roots of Polynomials with Integer Coefficients . . . . . . 1903.7.6 Sign of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1943.7.7 Sign of Rational Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

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QuizPolynomials and rational functions . . . . . . . . . . . . . . . . . . . . . . . . 209Self-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Answers to Self-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2144 Limits and Continuity of Functions 216

4.1 Definition of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2164.2 Properties of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2364.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

QuizDefinition and properties of limits, continuity . . . . . . . . . . . . . . . . . . 2444.4 Evaluation of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

4.5 Limits of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274QuizEvaluation of limits of functions and sequences . . . . . . . . . . . . . . . . . 276

5 Derivative 278

5.1 Definition of derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2785.2 Evaluating Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

QuizDefinition and evaluation of derivatives . . . . . . . . . . . . . . . . . . . . . 3165.3 Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3185.4 Tangent and Normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3215.5 Theorems on Continuous and Differentiable Functions . . . . . . . . . . . . . . . . . 326

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345QuizHigher order derivatives, tangent and normal, theorems on continuous and differ-

entiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34910
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Self-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351Answers to Self-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

6 Applications of Derivatives 353

6.1 LHospitals Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365QuizLHospitals Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

6.2 Monotonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3696.3 Local Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

QuizExtrema and Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3906.4 Convex and Concave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3956.5 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

QuizConvex and Concave Funcions, Asymptotes . . . . . . . . . . . . . . . . . . . 4196.6 Behaviour of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

6.7 Absolute Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457QuizAbsolute Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Self-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461Answers to Self-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

7 Approximation of Functions with Polynomials 464

7.1 Differential of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46411


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7.2 Taylor Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4747.3 Taylors Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

7.4 Maclaurins Formulae of Some Elementary Functions . . . . . . . . . . . . . . . . . . 490Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

Appendix 505

A.1 Basic Formulae for Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . 505A.2 Algebraic Equations of the Third and Fourth Degree . . . . . . . . . . . . . . . . . . 508

A.2.1 Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

A.2.2 Quartic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

Bibliography 515

Index 517

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13

Chapter 1

Basic Concepts

The goal of this chapter is to briefly reintroduce and complete the concepts and knowledge without whichwe would not be able to proceed with our study of differential calculus. First we will examine structureof mathematical texts and concepts from set theory. Then we will consider real numbers and explainthe basic difference between the set of real numbers and the set of rational numbers. Finally, we will

introduce the concept of mapping in preparation for the following chapter, devoted to functions.

1.1 Structure of Mathematical Texts

Since this is the first mathematical textbook which bachelor students encounter we will explain sometypical elements and features. Some features will be familiar from secondary school textbooks, whileothers are probably new. An ideal, strictly rigorous, mathematical theory should look as follows:

basic undefined conceptswith which we will work are named.


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Basic Concepts 14

axioms, statements about the properties of basic concepts assumed true and left unproven, are outlined. definitions, introduce new concepts built from basic and previously defined concepts. theorems, concerning the properties of defined concepts are formulated. These usually includeas-

sumptionsand a propositionor statementstating what is valid if the assumptions are satisfied. Themost common forms of propositions areimplication(e.g., if an integer is divisible by four, it is divisi-ble by two) and equivalence(e.g., an integer is divisible by six if and only if it is divisible by two andthree).Sometimes, instead of theorem the wordstatementis used. In some texts we encounter both; in whichcase the theorem is the more important result.

Inproofs, using the tools of mathematical logic it is verified that under the assumptions of a theorem itsproposition is valid. Each theorem must be proven, relative to the form we are discussing, for example,direct proof, indirect proof, proof by contradiction, and proof by mathematical induction.

lemmas, are statements created for the sake of simplicity when proofs are long and complicated. Alemma is an auxiliary statement which is used in proofs of other theorems.

We also encountercorollaries, statements easily implied by theorems,hypotheseswhich are unprovenstatements (we do not know if they are valid or not), exampleswhich illustrate the use of theorems,andcommentsorremarkswhich elaborate on the matter under consideration.

It is impossible to proceed in this manner in texts which are not designed for professional mathemati-cians, and it is even less viable for beginner texts.

In our exposition we will build on concepts and statements with concepts known from secondaryschool. Only theorems which are necessary and within the capacity of engineering students will beproven. On the other hand we will elaborate newly introduced concepts using examples of their appli-cations in physics and other technical subjects. We will concentrate on the proper understanding and


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Basic Concepts 15

visualization of new concepts. We will emphasize the exact formulation of theorems and understandingtheir content. Applications of theorems will be illustrated by examples solved in detail and accompaniedwith numerous figures.

1.2 Sets

The concept of set is one of the most important concepts of modern mathematics. Bysetwe understanda collection of objects. The particular objects which a given set contains are called its elements.

The notationaA means thata is an element of setA. The notationa /A means thata is not anelement of setA. We also say that elementa belongs or does not belong to set A.

If setA contains the elements a , b, and c, we put the elements into curly brackets and writeA== {a , b , c}. This formulation can be used if the set contains a small number of elements or if it is clearhow they are created. For example,B= {1, 2, 3, . . . , 8},C= {2, 4, 6, . . . }.

The most frequent way to describe sets is based on some characteristic property of their elements.The notationD= {x E: V (x)}means thatD contains just those elements from set E having thepropertyV (x). For example, setD= {x N: 3 x < 7} contains all natural numbers greater thenor equal to three and less than seven. Therefore,D= {3, 4, 5, 6}.

A set can also itself be an element of a set. To avoid problems we do not admit the existence of a setcontaining all sets because this leads to logical problems such as well-knownRussells1 paradox.

1Bertrand Arthur William Russell (18721970)British philosopher, logician, mathematician, historian, and socialcritic. In 1901 he popularly formulated his well-known paradox as follows (adopted from[10]):There was once a barber. Some say that he lived in Seville. Wherever he lived, all of the men in this town either shaved them-selves or were shaved by the barber. And the barber only shaved the men who did not shave themselves.That is a nice story. But it raises the question: Did the barber shave himself? Lets say that he did shave himself. But we seefrom the story that he shaved only the men in town who did not shave themselves. Therefore, he did not shave himself. But we

again see in the story that every man in town either shaved himself or was shaved by the barber. So he did shave himself. Wehave a contradiction.
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Set theoryis a mathematical discipline established by Cantor1 that studies general properties of sets.We will only use a very small part of the tools of the field, namely those for concise and exact notation.

Let A and Bbe sets. We say that thesets A andBare equaland write A=B if and only if any elementcontained inAis also contained in B and any element contained in B is also contained inA.

For example, ifA= {1, 2} andB= {2, 1}, thenA=B .If two setsAandB are not equal, we write A=B .

LetAandB be sets. We say that the setB is a subset of the setAand writeB A if and only if anyelement contained inB is also contained inA.

The aspect of being a subset, denoted by , is calledinclusion. The notation AB is consideredto be equivalent to B A. ThenA is also called a supersetofB . IfB is not a subset ofA, we writeBA.

For many purposes it is useful to introduce a set containing no element also called an empty set. Thisset is denoted by or {}.

Comment 1.1 The following properties are evident:

1) For any setAthere isAA.2) LetAandB be sets. Then A=B if and only ifAB and at the same timeBA. This property

is often used to prove the equality of two sets.

1Georg Ferdinand Ludwig Philipp Cantor(18451918) (read kantor)outstanding German mathematician, inventor ofset theory. One of the greatest mathematicians of all times David Hilbert (18621943) declared: No one shall expel us from

the Paradise that Cantor has created.


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Basic Concepts 17

3) LetA,B , andC be sets such thatAB and B C . ThenAC .Example 1.2 Find all the subsets of setA

= {1, 2, 3

}.

Solution. Ahas the following subsets:, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.


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Basic Concepts 18

Let us recall the basic set operations: union, intersection, difference, and complement.

1) Union of setsA andB (denotedA B) is the set of all elements contained in A orB .2) Intersection of setsA andB (denotedA B) is the set of all elements contained in A andB .3) Difference of setsA andB (denotedA B) is the set of all elements contained in Athat are not at

the same time contained inB .

4) Assume that A is a subset of some basic set Z. Thencomplement ofA with respect to Z(denoted A

or AZ) is the set of all elements inZthat are not contained inA.

Venn1 diagramsare often used to visualize sets and set operations:

A

B

Z

A B

A

B

Z

A B

A

B

Z

A B

A

Z

A

Similarly, the union and intersection of three or more sets is introduced. Numerous laws can bederived for set operations. We can only mention some. Using Venn diagrams try to draw figures cor-

1John Venn(18341923)English mathematician and logician.


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Basic Concepts 19

responding to left-hand and right-hand sides of the following laws. Equality is valid if the same figurescorrespond to both sides.

A B= B A, A B=B A commutative laws(A B) C=A (B C) associative law(A B) C=A (B C) associative law(A B) C=(A C) (B C) distributive law(A B) C=(A C) (B C) distributive law(A B)=A B , (A B) =A B De Morgans1 laws(A)=A, A B= A B

Example 1.3 LetA= {1, 2, 3, 4} andB= {2, 4, 5}. FindA B,A B,A B.Solution. We haveA B= {1, 2, 3, 4, 5},A B= {2, 4},A B= {1, 3}.

1Augustus De Morgan(18061871)British mathematician and logician.


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1.3 Number sets

Number setsare sets having numbers as elements. In mathematical analysis we will work almost exclu-

sively with number sets. Let us recall some learned in secondary school.

N = {1, 2, 3, . . . } the set of all natural numbersZ = {. . . , 2, 1, 0, 1, 2, 3, . . . } the set of all whole numbers or integersQ = p/q: p Z, q N the set of all rational numbersR the set of all real numbers

I = R Q the set of all irrational numbersC the set of all complex numbers

We also use the following notation:

R+= {x R : x >0} the set of all positive real numbers,R+0= {x R : x 0} the set of all positive real numbers including zero,

and similarly N0, R, R0, Q+, Q+0, Q, Q0, Z+, Z+0, Z, Z0.

Comment 1.4

a) The inclusions N Z Q R hold.b) Decimal representation of a rational number is either a terminating decimal(for example 3

4= 0.75)

or aperiodic decimal(for example 2311

= 2.09=2.090909 . . . or 3730

= 1.23=1.23333 . . . ).c) Decimal representationof an irrational numberis a non-periodic decimal (forexample =3.141592 . . . ,

2=1.414213 . . . , 3

5=1.709975 . . . ).


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1.3.1 Intervals

Later we will work intensively with special subsets of R known as intervals. We will summarize their

definitions and notation.

Definition 1.5 Leta, b R,a < b.i) A closed bounded intervalwith endpointsa andbis a set

a, b = {x R :a x b}.

ii) An open bounded intervalwith endpointsa andbis a set

(a, b)= {x R :a < x < b}.

iii) A left-closed and right-open bounded interval with endpointsa andbis a set

a,b)= {x R :a x < b}.

iv) A left-open and right-closed bounded interval with endpointsa andbis a set

(a,b = {x R :a < x b}.

v) A left-closed unbounded interval with endpointa is a set

a, +)= {x R :a x}.

vi) A right-closed unbounded intervalwith endpointbis a set

(, b = {x R :x b}.


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Basic Concepts 22

vii) A left-open unbounded intervalwith endpointa is a set

(a,

+)

= {x

R

:a < x

}.

viii) A right-open unbounded intervalwith endpointbis a set

(, b)= {x R :x < b}.

Likewise set R is considered to be an interval and the notation R=(, +)is used. Bounded andunbounded intervals are known, respectively, as finite and infinite intervals. Intervals closed on the leftand open on the right, or vice versa, are called half-open or half-closed.

Intervals can be classified from various points of view: bounded and unbounded, closed and open,and so on. In some textbooks a different notation is used:[a, b] instead ofa, b, ]a, b[ instead of(a,b)and analogously other types.

R, the set of all real numbers plays an important role in mathematics. The formal definition ofRis difficult and technically very complicated. It relies upon the consecutive and lengthy construction ofnatural numbers, integers and rational numbers, from which R is finally constructed. This approach isimpossible in a mathematics course for engineers. So for our purposes the geometric interpretation of

real numbers is sufficient. We identify real numbers with points on a straight line(number line or realline). Each real number corresponds to exactly one point on the number line and vice versa.

1.3.2 Properties of Real Numbers

Let us briefly recall a few basic properties of real numbers from secondary school.

Real numbers can be addedand multiplied.

To any real numbera there exists an opposite number

a.


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Basic Concepts 23

To any nonzero real numbera there exists an inverse numbera1 =1/a. Commutative, associative, and distributive laws are valid for addition and multiplication:

a+

b=

b+

a, (a+

b)+

c=

a+

(b+

c), a

(b+

c)=

a

b+

a

c,

a b=b a, (a b) c=a (b c), (a+ b) c=a c + b c. Subtraction can be defined as addition of the opposite number: a b=a+ (b). Division by a nonzero number can be defined as multiplication by the inverse number:a:b=a b1. Real numbers areorderedwith respect to their size.

But rational numbers have the same properties. Thus it is natural to ask why we prefer real numbers to rationalnumbers and what are the properties that real numbers have that rational do not. The reason is that there is a one-

to-one correspondence between real numbers and points of the number line; real numbers fully cover it. Rationalnumbers do not fully cover the number line, a lot of holes remain that correspond to irrational numbers. To explainthe difference more exactly we introduce several important and useful concepts.

Definition 1.6 LetM R andk, l R. We say thati) kis an upper bound of setMifx kfor anyxM.

ii) lis a lower bound of setMifx lfor anyxM.

iii) kis maximum of setMifk is an upper bound ofMand k M. Therefore,k is the greatest element ofM.We denotek=max M.iv) lis minimum of setMifl is a lower bound ofMand lM. Therefore,l is the smallest element ofM. We

denotel=min M.The setMis said to beupper bounded(lower bounded) if it has at least one upper (lower) bound. The set Missaid to beboundedif it is both upper and lower bounded.

Evidently, ifk is an upper bound ofM, then any number k > kis also an upper bound ofM, and similarly,

ifl is a lower bound ofM, then any numberl < lis also a lower bound ofM.


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Basic Concepts 24

Now we are prepared to define two important concepts tied in with real numbers.

Definition 1.7 LetM

R.

i) IfMis upper bounded and s is its smallest upper bound, then s is calledsupremumorthe least upper boundof setM(we denotes=sup M).

ii) IfMis lower bounded andi is its greatest lower bound, then i is called infimumorthe greatest lower boundof setM(we denotei=infM).

IfMhas the greatest element, it is at the same time the smallest upper bound and sup M= max Mholds.Analogously, ifMhas the smallest element, it is at the same time the greatest lower bound and infM= min Mholds. If this is not the case, it is not clear at first sight whether supremum or infimum must exist. That is why the

following theorem is amongst the key properties of real numbers and has fundamental importance in the rigorousconstructs of differential and integral calculus.

Theorem 1.8 Every upper bounded setM R has supremum and every lower bounded setM R has infimum.

The previous theorem is the rigorous formulation of the informal comment that there are no holes in R.The following example illustrates the difference between the concepts maximum and supremum and minimum

and infimum.

Example 1.9 Determine minimum, maximum, infimum, and supremum of the given subsets of R: A= 2, 7),B= {1, 2, 3},C= N,D= 3, +),E=(3, +), andF= R.Solution.

SetA:Ahas the smallest element2hencemin A=2=infA. The greatest element does not exist (there is no greatestnumber less then 7) hence max Adoes not exist. It is upper bounded, an upper bound is any numberu 7.Therefore, the smallest upper bound is7andsup A=7.SetB :

Bhas both the smallest and greatest element hence min B=1=infBand max B=3=sup B.


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SetC :Chas the smallest element1 hencemin C= 1=infC. It is not upper bounded hence neither max Cnor sup Cexist.

SetD:Dhas the smallest element 3hencemin D=3=infD. It is not upper bounded hence neither max Dnorsup Dexist.

SetE:Edoes not have the smallest element (there is no smallest number greater then 3) hencemin Edoes not exist. Itis lower bounded, a lower bound is any numberl 3. Therefore, the greatest lower bound is 3 andinfE= 3. Itis not upper bounded hence neither max Enorsup Eexist.

SetF:F is neither upper nor lower bounded hence none of numbers min F,infF,max F, andsup Fexist.

Proposition of Theorem1.8is not valid for the set Q of all rational numbers. For example, the setA== {x R:x 2 < 2} is a different notation of open interval(

2,

2 ). Therefore, in R there isinfA=

2

andsup A=

2. But the set B= {x Q: x2 < 2}(understood as a subset of Q) does not have supremumand infimum because

2are not rational numbers (this fact is usually proven at secondary school). There is no

smallest rational number among upper bounds ofB , i.e. rational numbers greater than

2, and analogously nogreatest rational number among lower bounds ofB , i.e. rational numbers smaller than

2. So there are gaps

in Q.Finally, let us note the fact that rational and irrational numbers are spread on the number line very densely.

Between any pair of different real numbers there are infinitely many rational and irrational numbers.

1.3.3 Powers and Roots of Real Numbers

The aim of this section is to provide a reasonable definition ofa b, wherea > 0and b are any real numbers. Wewill only sketch the main idea, further details can be found in [1].


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For anyn N anda R the symbolan means the abbreviation of the product ofncopies ofa , that isan =a a a

n-times.

Further, for anyn Z,n 0, and any n N there exists the unique b > 0 such that

a= bn. Therefore, b is the only positive solution of this equation. The numberb is called the n-th root ofaand denoted b= na. This fact is well-known from secondary school but the existence of roots could not beproven with the tools then available to you. Using the concepts introduced above, it is possible to verify that thesetM= {x R : x >0, xn a} is non-empty and upper bounded, its supremum b=sup Mhas the propertybn

=a , and is the only positive real number with this property.

IfnN is an even number anda < 0, the equationbn = a has no solution because bn 0 for any b R.But ifn is odd, there exists the unique solution b= n|a|. Hence, it is natural to extend the definition of oddroots this way also fora 0and r Q Z, r = p/q , where p Zand q N, q 2, that is, ifr is a rationalnumber which is not an integer, we define a r = ap/q = qap . This is the standard secondary school approach.The definition is correct, it does not depend on the representation ofr as the quotient of two integers (there areinfinitely many such quotients for eachr Q, e.g. 3

4= 6

8= 9

12= ). It is well-known that ifp/q= m/ nfor

somem, n, p, q , the equality q

ap = n

am holds.

B i C 27


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Basic Concepts 27

In some cases, the symbol a p/q , p Z, q N, can also be defined for a = 0or a < 0. If p/q > 0,we set 0p/q = 0. Ifa < 0and fraction p

qis in lowest terms (that is, pand q are coprime) and q is odd, we

seta p/q

= q

ap, which is in agreement with the fact that odd roots are also defined for negative numbers. For

example, (8)2/6 = (8)1/3 = 3(8)= 2is defined while(8)6/4 is not defined since the simplified formof 6

4 is 3

2and the denominator is even.

Finally, we are ready to define a b for any a, b R, a > 0. The idea is to approximateb R Q with arational numberr , to calculatear and accept this number as an approximation ofab. The nearer will berto b, thenearer will bear toab. Formally we proceed as follows:

Fora >1 we defineab =sup{ar :r Q, r b}.For0 < a 1from0 < a < 1is that for a > 1the number a r ,r Q, increaseswith increasingr while for0 < a < 1it decreases with increasingr . Thus in the first casear is always smallerthena b and approximate it from below while in the second case it is always greater then a b and approximate itfrom above.

In the previous definitions, we considered rational numbers r b. In its stead, it is possible to use rationalnumbersr b defining a b

= inf

{ar

: r

Q, r b

}for a > 1and a b

= sup

{ar

: r

Q, r b

}for

1< a


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Basic Concepts 28

only ifa = kn, wherek N, thus ifa is the n-th power of some natural number. This means that most rootsof natural numbers are irrational. For example,

2, 3

2,

5,

10, 4

3, 5

5, and 10

7are irrational. Therefore,

the set of all rational numbers does not even contain most roots of natural numbers. A similar conclusion is valid

for roots of all rational numbers and also for values at rational numbers of exponential, logarithmic, trigonometric,

and other functions that will be studied in Chapter3. All this explains why rational numbers are not sufficient for

our needs.

1.4 Extended Set of Real Numbers

For some purposes, namely the evaluation of limits, it is useful to extend set R by two elements

+and

, which lie intuitively before and after all real numbers, respectively. We obtain the extended set ofreal numbers R. Thus,

R = R {, +}.Some operations and ordering can be extended to R:First, ordering is defined in an expected way:

For anyx R : < x


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Basic Concepts 29

forx R+ {+}: x (+)= + x= +,x ()= x= ,

forx R {}: x (+)= + x= ,x ()= x= +,

forx R : x+ =x

=0,|| = |+| = +.

It is very important to realize which operations are not defined:

+ + (), + (+), 0 (), () 0, , x

0, x R.

Instead of the symbol +, the shorter form can be used (the sign + may be omitted). Analogouslyto Definitions1.6and1.7, the concepts upper and lower bound, maximum, minimum, supremum, andinfimum can be introduced. It is easy to verify that unlike the situation in R, any subset ofR has the

supremum and infimum.

Comment 1.10 The concept supremum (infimum) of subsets ofR was only introduced for upper (lower) boundedsets. As R is a subset ofR it is natural to generalize definitions of these concepts for unbounded subsets of R:

(a) For a nonempty upper unbounded setM R we definesup M= +(+ is the only upper bound ofMunderstood as a subset ofR).

(b) For a nonempty lower unbounded setM R we defineinfM= ( is the only lower bound ofMunderstood as a subset ofR).

Basic Concepts 30


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(c) For the empty set we defineinf = +,sup = . This definition is reasonable because any elementofR is both upper and lower bound of the empty set. Therefore, R is both the set of all upper bounds andthe set of all lower bounds of the empty set.

After this generalization any subset ofR has supremum and infimum. In Example1.9,of course, some answerswill change: sup C=sup D=sup E=sup F= +,infF= .

1.5 Mappings and Functions

A mapping or map is one of the most important mathematical concepts.

Definition 1.11 A mapping f of setA to setB is a rule which assigns the unique elementy B toeach elementx A. The notationf:AB is used.The elementy assigned to element x is denotedf (x), i.e. y= f(x), and is calledthe image ofx orthe value of mappingf atx . Likewise we use the notationxy .The set A is called the domain of mapping fand is denoted Dom(f ). A set of all elements y Bwhich are assigned to some x A is calledthe rangeorthe image of mapping fand is denotedIm(f ).The inclusionIm(f )

B holds. (The setB is called codomain of mappingf, but we will not need

this term.)

Iff:A B is a mapping and A and B are number sets (or at least B is a number set) instead ofmapping we use the termfunction.

We can imagine the domain A=Dom(f )as a set of inputs, the codomain B as the set of possibleoutputs, and the range Im(f ) as the set of real outputs. The output corresponding to the inputx isdenotedf (x).

Basic Concepts 31


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The wordruledoes not express any rigorously defined mathematical term. The formal definition ofmapping relies on the concepts of ordered pair, Cartesian product and relation. For our purposes, theprevious definition is sufficient.

Mappings or functions can be described in many ways. We will present a few examples.

1. A mapping f:A B can be visualized with an arrow diagram. Using Venn diagrams to displaysets A and Bwe relate any xA with the assigned f(x)B by a directed arrow. Therefore, a singlearrow starts at eachx A. ThoseyB at which at least one arrow ends represent the rangeIm(f ).For example, supposeA= {a , b , c}and B= {1, 2, 3}. In Fig.1.1two mappings f andg fromAtoB are displayed. ThenIm(f )= {1, 2} =B and Im(g)= {1, 2, 3} =B .

2. Another way is a word description. LetAbe the set of all triangles in the plane. Let us denote O(T )the area of a triangleT. ThenO : A R is a function assigning to the triangle T its areaO(T ).(We use the word function because O(T )is the number.)

3. The most common way of defining a function is with a formula. For example, f:R R given bythe formulaf (x)= x2 is a function assigning to x its second power x 2. Because the number x 2 isnonnegative every time and any nonnegative number can be obtained this way (e.g. 0= 02,4= 22or4=(2)2,5=(

5 )2 or5=(

5 )2 etc.), we haveIm(f )= 0, +).

4. Some functions are represented by a table. This is typical when the dependence of one numericalquantity on another is measured. An example of this type is function g : A R given by the table

x 2 0 1 3 5 8g(x) 0.25 2.1 1.33 2.52 1.38 2.94

Thus, for exampleg(2)= 0.25. For the domain and range we getDom(g)= {2, 0, 1, 3, 5, 8}andIm(g)= {2.52, 1.33, 0.25, 1.38, 2.1, 2.94}.

Basic Concepts 32
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ABa

b

c

1

2

3

f

ABa

b

c

1

2

3

g

Fig. 1.1

The symbols f andf(x)should be distinguished. The first is the notation of the whole rule, thesecond is the symbol of the element assigned to x. Sometimes, in less formal mathematical texts, thenotationf(x)is also used in the first meaning, namely if we want to indicate which letter is used for

elements of domainDom(f ).

Basic Concepts 33


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p

We will introduce three important types of mappings. Suppose f:AB is a mapping. IfIm(f )=B , thenfis calledsurjectionoronto mappingofAontoB . If for any pair x1, x2 A, x1= x2, also f (x1)= f (x2), thenf is called injectionor one-to-one

mappingofAintoB .Injection assigns different outputs to different inputs. In arrow diagram representation, arrows startingat different elements ofAare not allowed to end at the same element ofB .

Iff is both surjection and injection, it is calledbijectionorone-to-one onto mappingofAontoB .Each injection fwhich is not surjective can be formally changed to bijection if its codomain B isreduced toIm(f ), hence the elements fromB Im(f )are removed.

Having a look at Fig.1.1we can see thatf is neither the surjection nor injection whilegis both thesurjection and injection so it is bijection.

The remainder of this text will be devoted to the study of special functions calledreal functions of asingle variable. These will be introduced in the following chapter.

Exercises

1. Determine minimum, maximum, infimum, and supremum of the following subsets of R:

a) A=(0, 5) {7} (8, 9, b) B= {n! :n N} = {1!, 2!, 3!, . . . },c) C= Q= Q (, 0), d) D=(, 3) (5, +),e) E= {x R : x 2 >4}, f) F= {x R :6x x2 9< 0},g) G=(2, 2) (0, 4), h) H= (2, 2) 0, 4),i) I= (2, 2 (0, 4), j) J=

1n: n N

=

1, 1

2, 1

3, 1

4, . . .

,

k) K

=(

1, 0)

(0, 1), l) L

=(

1, 0

0, 1).

Basic Concepts 34


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p

2. Evaluate:

a) (1)2/6, b) (1)3/6, c) (32)2/10, d) (32)5/10,e) 9

3/6, f) (

2)

12/4, g) (

27)

4/12, h) (

4)12/8,

i) (4)8/12, j) (32)12/20, k) (243)12/15, l) (81)20/16.

3. Determine which of the following equalities are true:

a) (1)2/6 = 6

(1)2, b) (32)3/15 = 15

(32)3, c) (2)2/4 = 4

(2)2,d) (1)8/4 = 4

(1)8, e) (32)12/20 = 20

(32)12, f) (2)4/6 = 6

(2)4.

Solutions1. a) min Adoes not exist, max A=9, infA=0, sup A=9,

c) min B=1, max Bdoes not exist, infB= 1, sup B= +,b) min Cdoes not exist, max Cdoes not exist, infC= , sup C=0,d) min Ddoes not exist, max Ddoes not exist, infD= , sup D= +,e) min Edoes not exist, max Edoes not exist, infE= , sup E= +,f) min Fdoes not exist, max Fdoes not exist, infF

= , sup F

= +,

g) min Gdoes not exist, max Gdoes not exist, infG=0, sup G=2,h) min H= 0, max Hdoes not exist, infH= 0, sup H= 2,i) min Idoes not exist, max I= 2, infI= 0, sup I= 2,

j) min Jdoes not exist, max J= 1, infJ= 0, sup J= 1,k) min Kdoes not exist, max Kdoes not exist, infK= +, sup K= ,l) min L=0, max L=0, infL=0, sup L=0.

Basic Concepts 35


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2. a) 1, b) undef., c) 1/2, d) undef., e) 1/3, f) 1/8,g) 1/3, h) undef., i) 2 3

2, j) 8, k) 81, l) undef.

3. a) 1=1, b) 2= 2, c) undef.= 2,d) 1=1, e) 8=8, f) 3

4= 3

4.

QuizBasic concepts

Choose the correct answer (only one is correct).

1.(1 pt.) For every setAthere is A.Yes. No.

2.(1 pt.) For every setAthere is A.Yes. No.

3.(1 pt.) Which of these sets does not contain3.1415?

Q R I C

4.(1 pt.) Which of these sets is empty?

(1, 3) (2, 4) (1, 2) (3, 4)

1, 2 2, 4 1, 3) (3, 4

Basic Concepts 36


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5.(1pt.)

1

2

12

=

22

12

2 4

6.(1pt.) 27

27=3

3 9

3

3

9 9

9

7.(1 pt.) The set(, ) 5, + is bounded.Yes. No.

8.(1 pt.) Empty subset ofR is bounded.Yes. No.

9.(1 pt.) Minimum of the set(1, 2) {3} (0, +)is:0 1 Does not exist.

10.(1 pt.) Supremum of the set(1, 2) {3} (0, +)in R is:2 3 + Does not exist.

Correct Answers:Points Gained:

Success Rate:

37


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Chapter 2

Real Functions of a Single Variable

2.1 Definition of a Function and its Graph

Below are descriptions of material covered in the remaining text.

Definition 2.1 SupposeA

R, A=

. A mappingf ofA into R(f:

A

R) is called a real

function of a single variable(hereafter known as thefunction).

Therefore, functionf is a rule assigning a unique real number to each number fromA. The numberx Ais often called the independent variableand the numberf (x)related to it is called the dependentvariable. This terminology expresses the fact that the valuef (x)depends on the choice ofx . Let usremember thatA is the domain of fand is denoted Dom(f )and the set of all values f (x)whichfassumes is called the image or range offand is denotedIm(f ).

Real Functions of a Single Variable 38


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2.1.1 Definition of Functions

Defining function fwe must describe its domain Dom(f ) and the rule assigning a unique number

y Im(f )to each x Dom(f ). As we have already mentioned functions can be represented by aformula, table, arrow diagram, and word description.The most frequent way is through a formula. For example, a functionfwhich assigns to each

x 2, 5 =Dom(f )the numberx 2 + 1can be written as follows:

f: y=x 2 + 1, x 2, 5,f(x)=x 2 + 1, x 2, 5,

f: xx2

+ 1, x 2, 5.In many instances only a formula is given and the domain is not explicitly mentioned. In cases like

this the set of allx R where it makes sense in the formula is regarded as the domain. We call this setthe natural domain.

2.1.2 Equality of Functions

The two functions f andg are regarded as equal (in writing f= g) if and only if they have the samedomain, and for anyx of this domainf (x)=g(x)holds.Example 2.2 Determine if functions

f: y=x+ 1, x(, 0), and g : y= x2 1

x 1 , x(, 0)

are equal.



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Solution. The domains are the same and for anyx(, 0)we havex2 1x 1 =

(x+ 1)(x 1)x 1 =

x

+1,

sof (x)=g(x). Therefore,f=g . Example 2.3 Determine if functions

f: y= 2 ln x and g : y=ln x2

are equal.

Solution. This time only the formulae are given and we have to determine their natural domains:

Dom(f )= {x R : 2 ln xhas the sense} = {x R : x >0} =(0, +),Dom(g)= {x R : ln x2 has the sense} = {x R : x2 >0} = R {0}.

The domains Dom(f )=(0, +) and Dom(g)= R {0} =(, 0)(0, +) are different, hencef=g .

Note that in the common part of the domains Dom(f )

Dom(g)=

(0,+

)both formulae givethe same value becauseln x2 =2 ln xforx >0.

2.1.3 Graph of a Function

We will illustrate another method by which a function can be visualized.Let us remember that the symbol R2 indicates the set of all pairs of real numbers. These pairs can

be regarded as Cartesian coordinates of points on a plane. Hence pairs of real numbers can be identified

with points on a plane and this plane with the set R2

.



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Iff: Dom(f ) R is a function, the pair[x , f ( x )]can be understood as the coordinates of thepoint, which we may display in the plane. Doing this for any numberx Dom(f )we obtain the set ofpoints in the plane called the graph off. The formal definition is:

Definition 2.4 The graph of functionf: Dom(f ) R is a set of points

G= {[x, y] R2 :xDom(f )andy=f (x)}.

Thus the graph of functionfis the set of points on a plane, the first coordinate of which (abscissa)is the numberxDom(f ), and the second coordinate (ordinate) is the corresponding valuey= f (x).

Generally, not every set of points in the plane is the graph of a function. If the set G

R2 represents

the graph, then it does not involve the points[x, y1],[x, y2], wherey1= y2, that is, those points lyingone above the other. It is the consequence of the definition of functionthe uniquey is assigned to anyxDom(f ).

A set of points G on a plane is the graph of function fif and only if any straight line parallel to they-axis intersectsGat no more than one point.

The situation is illustrated in the following figures.



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x

y

The circle is not the graph of a function

x

y

The parabola is the graph of a function

Example 2.5 Draw the graph of the function

f:y= 1, x 0.

x

y

O

1

1

y=sgn x

Solution.The graph consists of two open half-lines and an isolated point.This function is calledsignand is denotedsgn, i.e. f:y= sgn x. Evi-dently,Dom(f )= R andIm(f )= {1, 0, 1}.



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Example 2.6 Draw the graph of the function

f

:y

= x, x


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5. ifa >0, then |x| aif and only ifx a, a and|x|< aif and only ifx(a,a).

The graph concept is very important and useful. Considerable information about the properties ofa function can be identified from a graph. For example, we can determine the domain and range of afunction. The domain is obtained as the orthogonal projection of the graph to thex-axis and the range isobtained as the orthogonal projection of the graph to the y-axis. Likewise, functions are often representedby their graphs.

Comment 2.8 Previous examples may imply it is possible to draw the graph of any single variable function. Butthis conclusion is false. Consider, for example, the function

: y= 1, x Q,0, x I.

This function assumes only two values, namely zero and one, depending on x Qor x I. Thus (1)== ( 3

2)= ( 18

17)= 1, ()= (

2)= 0etc. HenceDom( )= R andIm( )= {0, 1}. Because rational

and irrational numbers are interspersed very densely on the number line, the graph of is broken into individualpoints, contains no connected arcs and cannot be reasonably plotted. This function is called theDirichlet1 functionand is usually denoted by the small Greek letter .

2.2 Properties of Functions

In this section some basic properties of single variable functions will be examined.

1Johann Peter Gustav Lejeune Dirichlet (18051859) (read dirikle)important German mathematician. He was in-volved in number theory, mathematical analysis, and equations of mathematical physics.



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2.2.1 Bounded Functions

The first group of properties concerns the boundedness of the range.

Definition 2.9

i) Function f issaidtobe upperbounded on the setM Dom(f ) ifthesetofvalues {f(x):x M}is upper bounded.

ii) Function fis said to be lower bounded on the setM Dom(f ) if the set of values {f(x):x M}is lower bounded.

iii) Functionfis said to be bounded on the setM Dom(f )if it is both upper and lower boundedonM.

Functionfis said to be bounded(upper bounded, lower bounded) if it is bounded (upper bounded,lower bounded) onDom(f ).

Comment 2.10

1. It is clear from the previous definition that function f is upper bounded if and only if a constant L R

can be found such thatf (x) Lfor anyx Dom(f ). Thus the graph offlies under the straightliney= L (see Fig.2.1 a)).2. Similarly, function f is lower bounded if and only if a constant K R can be found such that

f(x) Kfor anyxDom(f ). The graph off lies over the straight liney=K (see Fig.2.1 b)).3. Finally, function f is bounded if and only if constants K, L R can be found such that K f(x) Lfor anyx Dom(f ). The graph offlies between the parallel lines y= Kandy= L(seeFig.2.1 c)).



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x

y

O

Ly=L

y=f (x)a) Graph of an upper bounded function

x

y

O

Ky=K

y=f (x)

b) Graph of a lower bounded function

x

y

O

L

K

y=L

y=K

y=f (x)

c) Graph of a bounded function

Fig. 2.1: Boundedness of functions



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4. If functionfis bounded, then it is bounded on any subset M Dom(f ). But if it is bounded onsome subset ofDom(f ), it need not be bounded. For example, the functionf(x)= x with thenatural domainDom(f )

=R is unbounded because its graph is the line of symmetry of the first and

third quadrants. But it is bounded on any bounded intervala, b. The same conclusion holds forother types of boundedness.

Let us emphasize once more that we speak about the boundedness of function values, not of the domain!

Example 2.11 Decide if function f (x)= x2 1

x2 + 1 ,x R, is bounded.

Solution. Arranging the formula we see that for anyx R

x2 1x2 + 1 =

x2 + 1 2x2 + 1 =1 +

2x2 + 1

holds. Asx 2 0, hencex 2 + 1 1for anyx R we get

0 1

x2 + 1 1 0 2x2 + 1 2 1

2x2 + 1 + 1 1.

Therefore,fis bounded (compare Fig.2.6;the middle graph corresponds to

f).

Example 2.12 Decide if functionf (x)= x1 + x2, x R, is bounded.

Solution. This time it is a bit more difficult to verify thatfis bounded. For any x R we have(|x| 1)2 0. Expanding we get |x|2 2|x| + 1 0, hence |x|2 + 1 2|x|. As |x|2 = x 2 dividing thelast inequality by the positive expression2(x2 + 1)(which does not reverse the inequality) we obtain

1

2

|x|x

2

+ 1.



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Using laws of absolute values (see Comment2.7) we have

|x|

x2

+ 1 =

|x|

|x2

+ 1| = x

x2

+ 1 = |f(x)| ,therefore we obtain

|f(x)| 12

which is equivalent to 12 f(x)

1

2.

This means thatf is bounded (compare Fig.2.6 a)).



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2.2.2 Monotonic Functions

The second set of properties concerns the behaviour of values of the function when we move along thegraph from left to right. We will study if function values increase or decrease.

Definition 2.13 Functionfis said to be:

i) increasing on the setM Dom(f )if for any x1, x2 M such that x1 < x2 the inequalityf (x1) < f (x2)holds,

ii) decreasing on the setM Dom(f )if for any x1, x2 M such that x1 < x2 the inequalityf (x1) > f (x2)holds,

iii) non-decreasing on the setM Dom(f )if for any x1, x2 Msuch thatx1 < x2the inequalityf (x1) f (x2)holds,

iv) non-increasing on the setM Dom(f )if for anyx1, x2 Msuch thatx1 < x2the inequalityf (x1) f (x2)holds, and

v) increasing, decreasing, non-decreasing, non-increasing,respectively if it is such on the whole do-mainDom(f ).

Functions that are increasing, decreasing, non-decreasing, or non-increasing are calledmonotonic.

Functions that are increasing or decreasing are called strictly monotonic.

Clearly, any increasing function is also non-decreasing and any decreasing function is also non-increasing. But the converse statements are not correct (monotonic functions which are not strictlymonotonic can be constant on some interval). The situation is displayed in Fig.2.2

We have not yet learned the appropriate tools to verify monotonicity (they rely on the derivative,which will be introduced in Chapter5)that is why we will only present a few very simple examples.



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x

y

O x1 x2

f (x1)

f (x2)

Graph of an increasing function

x

y

O x1 x2

f (x1)=f (x2)

Graph of a non-decreasing function

(which is not increasing)

x

y

O x1 x2

f (x1)

f (x2)

Graph of a decreasing function

x

y

O x1 x2

f (x1)=f (x2)

Graph of a non-increasing function(which is not decreasing)

Fig. 2.2



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Example 2.14 Identify the monotonicity of the following functions:

a) f(x)=x 2, x 0, +), b) g(x)=x 2, x (, 0,c) h(x)

=x 2, x

R, d) k(x)

=1/x, x

(0,

+),

e) l(x)=1/x, x (, 0), f) m(x)=1/x, x R {0}.Solution.

a) Assumex1, x2 0, +),x1 < x2. Then (becausex1,x2are nonnegative) we have x 21 < x22 , i.e.f (x1) < f (x2), therefore,fis increasing.

b) Assumex1, x2(, 0,x1 < x2. Thenx 21 > x22 , i.e. g(x1) > g(x2), thereforeg is decreasing.c) From the previous results we know that h is decreasing on (, 0 and increasing on 0, +). This

implies thathis not monotonic on(

,+

). The graph ofhis a parabola.

x

y

O 3 6

9

36

y=f (x)

x

y

O3366

9

36

y=g(x)

x

y

O

y=h(x)

Fig. 2.3



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d) Assume x1, x2 (0, +), x1 < x2. Then1/x1 > 1/x2, i.e. k(x1) > k ( x2), therefore k isdecreasing.

e) Assume x1, x2(, 0), x1 < x2. Then 1/x1 >1/x2, i.e.l(x1) > l(x2), therefore lis decreasing.

x

y

O 1 3

11/3

y

=k(x)

x

y

O1133

11/3

y=l(x)

Fig. 2.4

f) Due to the previous results we know that function m is decreasing both on (, 0)and (0, +)(we consider each interval separately). But it is not decreasing on the union (, 0)(0, +). Forexample, the pair3and3does not satisfy the condition from the definition of decreasing functionbecause 3< 3, but 1/3< 1/3. The graph ofmis a rectangular hyperbola (see Fig.2.4).



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x

y

333

A

1/3 1/3

y=m(x)

px

Fig. 2.5



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2.2.3 One-to-one Functions

On page33we introduced a special mapping called one-to-one. Since this concept will be important inthe definition of inverse functions we will formulate it once more for functions of a single variable, and

discuss it more thoroughly.

Definition 2.15 Functionf is called one-to-oneif for any pair x1, x2 Dom(f ), x1= x2, there isf (x1)=f (x2).

Thus each function value is assumed at a unique point. Verifying this property we can use an equiv-alent condition: If forx1, x2Dom(f )there isf (x1)=f (x2), thenx1=x2.

This property is easily seen on the graph:

Functionfis one-to-one if and only if any straight line parallel to the x-axis intersects the graph offat no more than one point.

Note thatany strictly monotonic function is one-to-one (function values at different points cannot bethe same). Buta one-to-one function need not be strictly monotonicas the function m in Example2.14shows. We learned that this function is not monotonic. Nevertheless, any line parallel to thex-axisintersects its graph at most once. The x-axis has no intersection point, all the other parallels intersect thegraph at a unique point (see point Ain Fig.2.5). Therefore,mis a one-to-one function.



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Example 2.16 Verify that functionf (x)=(x 1)2 + 7, wherex 1, +), is one-to-one.Solution. Assuming that for somex1, x2Dom(f )= 1, +)there isf (x1)=f (x2)we get

(x1 1)2 + 7=(x2 1)2 + 7,(x1 1)2 =(x2 1)2,|x1 1| = |x2 1|, (because

a2 = |a| for anya R)

x1 1=x2 1, (becausex1 1 0,x2 1 0)x1=x2.

Therefore,fis a one-to-one function. Be careful! A function with the same formula and a different domain need not be one-to-one. For

example,g(x)=(x 1)2 + 7, x R, is not one-to-one because forx1=0,x2=2there isx1=x2,butf (x1)=8=f (x2).

2.2.4 Even and Odd Functions

The following two properties concern the symmetry of the graph. We will consider functionsfwith thedomain symmetrical about the origin, that is with each x the domainDom(f )contains alsox. Thenthe comparison of function valuesf (x)andf (x) is meaningful.

Definition 2.17 Functionfwith a symmetrical domain is called;

i) eveniff (x)=f (x)for anyx Dom(f ), andii) oddiff (x)= f(x)for anyxDom(f ).



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Geometrically, the definition expresses the fact that;

i) the graph of an even function is symmetrical about the y-axis, andii) the graph of an odd function is symmetrical about the origin.

In general the function need not be even or odd. Thefunction f(x)=0 is both even andodd(this functionis the only one with this property because the conditions f (x)= f(x)andf (x)= f(x)implythatf (x)= f(x), i.e. 2f(x)=0, and sof (x)=0for anyx Dom(f )).Example 2.18 Decide if the following functions are even or odd.

a)f (x)= xx2

+1

, b)g(x)= 1 x2

1

+x2

, c)h(x)= 1 + x1

+x2

.

Solution. As the domains are not given we will consider the natural ones which, in all three cases, arethe set of all real numbers R.

Letx R. Replacing all occurrences ofx by xin each formula we get:a) f (x)= x

(x)2 + 1 =x

x2 + 1 = x

x2 + 1 = f(x). Hencefis an odd function.

b) g(x)= 1 (x)2

1

+(

x)2

= 1 x2

1

+x2

= g(x). Hencegis an even function.

c) h(x) = 1 + (x)1 + (x)2 =

1 x1 + x2 , which evidently seems different from both h(x)=

1+x1+x2 and

h(x)= 1x1+x2 . To prove it correctly we choose e.g. x= 1. Then h(1)= 1and h(1)= 0,

thereforeh(1)=h(1)andh(1)= h(1)andhis neither an even nor an odd function.To understand this better the graphs off,g, andhare displayed in Fig.2.6.



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x

y

O 1

1

1

f(x)= xx2 + 1

1/2

1/2

a)

x

y

O

g(x)= 1 x2

1+

x2

1

1

1 1

b)

x

y

O

h(x)= 1 + x1 + x2

1

1

c)

Fig. 2.6

http://prevpage/http://lastpage/http://goback/http://close/http://fullscreen/http://fullscreen/http://fullscreen/http://fullscreen/http://close/http://goback/http://lastpage/http://prevpage/


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2.2.5 Periodic Functions

Letfbe a function andp > 0 a real number. We will assume that with eachx Dom(f )the domainDom(f )also includes the numberx

+p. It then must also include the number(x

+p)

+p

=x

+2p,

(x+ 2p) + p=x+ 3pand so on. See the following figure.

x x+ p x+ 2p x+ 3p x+ 4p

Definition 2.19 Functionfis said to be periodicwith periodp,p R,p >0, if,i) for anyx Dom(f )alsox+ pDom(f ), and

ii) f (x

+p)

=f (x)for anyx

Dom(f ).

In other words, function values at points having the distance pare equal. Therefore, it is sufficient toknow the graph offon an interval of lengthp. The whole graph is obtained by copying this part whichis shifted bypto the right, or to the left if the domain admits it.

If a function is periodic with period p, any integer multiple k p,k N, is also the period. If thesmallest period exists it is called theprimitive period.

The best-known periodic functions are trigonometric functions. For example, sine and cosine havethe primitive period2 and functionf (x)

=sin 3xhas the primitive period2/3. In general, function

f(x)= sin ax , a > 0, has the primitive period 2/a. Functionf (x)= c, c R, is periodic. Itsperiod is any positive number. Therefore, it does not have the primitive period.

Example 2.20 Draw the graph of periodic functionfhaving the periodp=2and defined on R if youknow that

f(x)=

1 for x(1, 0),0 for x= 1andx=0,1 for x

(0, 1).



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Solution. Asfhas the period p= 2 it is sufficient to draw the graph offon the interval 1, 1)andthen to copy it to the right and to the left, each time after the shift 2. Notice thatf (1)cannot be definedarbitrarily because the equalityf (1)=f (1)must hold (distance of1and 1is just2). The result isdisplayed in Fig.2.7. Similar functions, known as periodic signals are often encountered in digital signal processing.

x

y

33 11 1 3 5O

1

1

Fig. 2.7

2.3 Operations with FunctionsWe will briefly discuss how basic operations with functions as the sum, difference, product, quotient,absolute value, and composition are defined. Greater attention will be paid to inverse functions and thecomposition of mutually inverse functions.



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2.3.1 Sums, Differences, Products, and Quotients of Functions

The definition below is natural. We add, subtract, multiply, and divide function values of two functionsat points where both these functions are defined. Moreover, in the case of division, the divisor must be

nonzero.

Definition 2.21 Suppose that fand gare functions. Theirsumf+g,difference fg,productfg,andquotientf /gare defined as follows:

(f+ g)(x)=f (x) + g(x) forx Dom(f ) Dom(g),(f g)(x)=f (x) g(x) forx Dom(f ) Dom(g),

(f g)(x)=f (x) g(x) forx Dom(f ) Dom(g),f

g

(x)= f(x)

g(x)forx Dom(f ) Dom(g) {z R :g(z)=0}.

The absolute value offis a function denoted |f| and defined by the formula|f|(x)= |f(x)| forx Dom(f ).

Instead off

gwe usually writef g. A special case of product of two functions is the multiplication by

a constant function. Forc R we define:(cf )(x)=c f (x), xDom(f ).

2.3.2 Composition of Functions

Consider two functionsf: Dom(f )R andg : Dom(g)R. If for somex Dom(f )the valuey

=f (x) is inDom(g), we can evaluate gat this number and obtain the value z

=g(y)

Im(g). This



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way we can construct a newcomposite functionassigning to some x Dom(f )the valuez Im(g),wherez=g(y)=g[f(x)].

This function is called the composition off andg and is denoted g f. It is defined for thosexDom(f )for whichf (x)Dom(g).In mathematics the symbol g f is read gcomposed with f. First we evaluate f(at the number x)and theng(at the numberf (x)). Therefore,

(g f)(x)=g[f(x)], x Dom(g f ),

whereDom(g f )= {x Dom(f ):f (x)Dom(g)}.

Function fis calledthe inner componentand function gthe outer componentof composite functiong f. The definition is presented visually in Fig.2.8using an

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