Real Analysis: Foundations and Functions of One Variable

Description

Based on courses given at Eotvos Lorand University (Hungary) over the past 30 years, this introductory textbook develops the central concepts of the analysis of functions of one variable -- systematically, with many examples and illustrations, and in a manner that builds upon, and sharpens, the student's mathematical intuition. The book provides a solid grounding in the basics of logic and proofs, sets, and real numbers, in preparation for a study of the main topics: limits, continuity, rational functions and transcendental functions, differentiation, and integration. Numerous applications to other areas of mathematics, and to physics, are given, thereby demonstrating the practical scope and power of the theoretical concepts treated. In the spirit of learning-by-doing, Real Analysis includes more than 500 engaging exercises for the student keen on mastering the basics of analysis. The wealth of material, and modular organization, of the book make it adaptable as a textbook for courses of various levels; the hints and solutions provided for the more challenging exercises make it ideal for independent study. Preface Contents 1 A Brief Historical Introduction 2 Basic Concepts 2.1 A Few Words About Mathematics in General 2.2 Basic Concepts in Logic 2.3 Proof Techniques 2.4 Sets, Functions, Sequences 3 Real Numbers 3.1 Decimal Expansions: The Real Line 3.2 Bounded Sets 3.3 Exponentiation 3.4 First Appendix: Consequences of the Field Axioms 3.5 Second Appendix: Consequences of the Order Axioms 4 Infinite Sequences I 4.1 Convergent and Divergent Sequences 4.2 Sequences That Tend to Infinity 4.3 Uniqueness of Limit 4.4 Limits of Some Specific Sequences 5 Infinite Sequences II 5.1 Basic Properties of Limits 5.2 Limits and Inequalities 5.3 Limits and Operations 5.4 Applications 6 Infinite Sequences III 6.1 Monotone Sequences 6.2 The Bolzano–Weierstrass Theorem and Cauchy's Criterion 7 Rudiments of Infinite Series 8 Countable Sets 9 Real-Valued Functions of One Real Variable 9.1 Functions and Graphs 9.2 Global Properties of Real Functions 9.3 Appendix: Basics of Coordinate Geometry 10 Continuity and Limits of Functions 10.1 Limits of Functions 10.2 The Transference Principle 10.3 Limits and Operations 10.4 Continuous Functions in Closed and Bounded Intervals 10.5 Uniform Continuity 10.6 Monotonicity and Continuity 10.7 Convexity and Continuity 10.8 Arc Lengths of Graphs of Functions 10.9 Appendix: Proof of Theorem 10.81 11 Various Important Classes of Functions (Elementary Functions) 11.1 Polynomials and Rational Functions 11.2 Exponential and Power Functions 11.3 Logarithmic Functions 11.4 Trigonometric Functions 11.5 The Inverse Trigonometric Functions 11.6 Hyperbolic Functions and Their Inverses 11.7 First Appendix: Proof of the Addition Formulas 11.8 Second Appendix: A Few Words on Complex Numbers 12 Differentiation 12.1 The Definition of Differentiability 12.2 Differentiation Rules and Derivatives of the ElementaryFunctions 12.3 Higher-Order Derivatives 12.4 Linking the Derivative and Local Properties 12.5 Intermediate Value Theorems 12.6 Investigation of Differentiable Functions 13 Applications of Differentiation 13.1 L'Hôpital's Rule 13.2 Polynomial Approximation 13.3 The Indefinite Integral 13.4 Differential Equations 13.5 The Catenary 13.6 Properties of Derivative Functions 13.7 First Appendix: Proof of Theorem 13.20 13.8 Second Appendix: On the Definition of Trigonometric Functions Again 14 The Definite Integral 14.1 Problems Leading to the Definition of the Definite Integral 14.2 The Definition of the Definite Integral 14.3 Necessary and Sufficient Conditions for Integrability 14.4 Integrability of Continuous Functions and Monotone Functions 14.5 Integrability and Operations 14.6 Further Theorems Regarding the Integrability of Functions and the Value of the Integral 14.7 Inequalities for Values of Integrals 15 Integration 15.1 The Link Between Integration and Differentiation 15.2 Integration by Parts 15.3 Integration by Substitution 15.4 Integrals of Elementary Functions 15.4.1 Rational Functions 15.4.2 Integrals Containing Roots 15.4.3 Rational Functions of ex 15.4.4 Trigonometric Functions 15.5 Nonelementary Integrals of Elementary Functions 15.6 Appendix: Integration by Substitution for Definite Integrals (Proof of Theorem 15.22) 16 Applications of Integration 16.1 The General Concept of Area and Volume 16.2 Computing Area 16.3 Computing Volume 16.4 Computing Arc Length 16.5 Polar Coordinates 16.6 The Surface Area of a Surface of Revolution 16.7 Appendix: Proof of Theorem 16.20 17 Functions of Bounded Variation 18 The Stieltjes Integral 19 The Improper Integral 19.1 The Definition and Computation of Improper Integrals 19.2 The Convergence of Improper Integrals 19.3 Appendix: Proof of Theorem 17timphely Erratum Hints, Solutions Notation References Index