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The second book, in whichwe also discuss the integral calculus of functions of several variables up to the general Newton–Leibniz–Stokes formula thus acquires a certain unity.We shall give more complete information on the second book in its preface. Atthis point we add only that, in addition to the material already mentioned, it containsinformation on series of functions (power series and Fourier series included), on integrals depending on a parameter (including the fundamental solution, convolution,and the Fourier transform), and also on asymptotic expansions (which are usuallyabsent or insufficiently presented in textbooks).We now discuss a few particular problems.On the Introduction I have not written an introductory survey of the subject,since the majority of beginning students already have a preliminary idea of differential and integral calculus and their applications from high school, and I could hardlyclaim to write an even more introductory survey.
Instead, in the first two chaptersI bring the former high-school student’s understanding of sets, functions, the useof logical symbolism, and the theory of a real number to a certain mathematicalcompleteness.This material belongs to the formal foundations of analysis and is aimed primarily at the mathematics major, who may at some time wish to trace the logicalstructure of the basic concepts and principles used in classical analysis. Mathematical analysis proper begins in the third chapter, so that the reader who wishes toget effective machinery in his hands as quickly as possible and see its applicationscan in general begin a first reading with Chap.
3, turning to the earlier pages whenever something seems nonobvious or raises a question which hopefully I also havethought of and answered in the early chapters.On the Division of Material The material of the two books is divided into chapters numbered continuously. The sections are numbered within each chapter sepa3 The “stronger” integrals, as is well known, require fussier set-theoretic considerations, outside themainstream of the textbook, while adding hardly anything to the effective machinery of analysis,mastery of which should be the first priority.xiiPrefacesrately; subsections of a section are numbered only within that section.
Theorems,propositions, lemmas, definitions, and examples are written in italics for greaterlogical clarity, and numbered for convenience within each section.On the Supplementary Material Several chapters of the book are written as anatural extension of classical analysis. These are, on the one hand, Chaps. 1 and 2mentioned above, which are devoted to its formal mathematical foundations, and onthe other hand, Chaps. 9, 10, and 15 of the second part, which give the modern viewof the theory of continuity, differential and integral calculus, and finally Chap. 19,which is devoted to certain effective asymptotic methods of analysis.The question as to which part of the material of these chapters should be includedin a lecture course depends on the audience and can be decided by the lecturer, butcertain fundamental concepts introduced here are usually present in any expositionof the subject to mathematicians.In conclusion, I would like to thank those whose friendly and competent professional aid has been valuable and useful to me during the work on this book.The proposed course was quite detailed, and in many of its aspects it was coordinated with subsequent modern university mathematics courses – such as, forexample, differential equations, differential geometry, the theory of functions ofa complex variable, and functional analysis.
In this regard my contacts and discussions with V.I. Arnol’d and the especially numerous ones with S.P. Novikovduring our joint work with the so-called “experimental student group in naturalscience/mathematical education” in the Department of Mathematics at MSU, werevery useful to me.I received much advice from N.V. Efimov, chair of the Section of Mathematical Analysis in the Department of Mechanics and Mathematics at Moscow StateUniversity.I am also grateful to colleagues in the department and the section for remarks onthe mimeographed edition of my lectures.Student transcripts of my recent lectures which were made available to me werevaluable during the work on this book, and I am grateful to their owners.I am deeply grateful to the official reviewers L.D. Kudryavtsev, V.P. Petrenko,and S.B.
Stechkin for constructive comments, most of which were taken into account in the book now offered to the reader.Moscow, Russia1980V. ZorichContents12Some General Mathematical Concepts and Notation . . . . . . . .1.1 Logical Symbolism . . . . . . . . . . . . . . . . . . . . . . . .1.1.1 Connectives and Brackets .
. . . . . . . . . . . . . . . .1.1.2 Remarks on Proofs . . . . . . . . . . . . . . . . . . . . .1.1.3 Some Special Notation . . . . . . . . . . . . . . . . . .1.1.4 Concluding Remarks . . . . . . . . . . . . . . . . . . .1.1.5 Exercises . . . . . . . . . . . . . . . . . . . .
. . . . . .1.2 Sets and Elementary Operations on Them . . . . . . . . . . . . .1.2.1 The Concept of a Set . . . . . . . . . . . . . . . . . . . .1.2.2 The Inclusion Relation . . . . . . . . . . . . . . . . . . .1.2.3 Elementary Operations on Sets . . . . . . . . . . . . . .1.2.4 Exercises . . . . . . . . . . . . . . . . . . . . .
. . . . .1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.1 The Concept of a Function (Mapping) . . . . . . . . . .1.3.2 Elementary Classification of Mappings . . . . . . . . . .1.3.3 Composition of Functions and Mutually Inverse Mappings1.3.4 Functions as Relations. The Graph of a Function . .
. . .1.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .1.4 Supplementary Material . . . . . . . . . . . . . . . . . . . . . .1.4.1 The Cardinality of a Set (Cardinal Numbers) . . . . . . .1.4.2 Axioms for Set Theory . . . . . . . . .
. . . . . . . . .1.4.3 Remarks on the Structure of Mathematical Propositionsand Their Expression in the Language of Set Theory . . .1.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .1112334557811121216171922252527The Real Numbers . . . . . . . . . . . . . . . . . . . . .
. . . .2.1 The Axiom System and Some General Properties of the Setof Real Numbers . . . . . . . . . . . . . . . . . . . . . . . .2.1.1 Definition of the Set of Real Numbers . . . . . . . .2.1.2 Some General Algebraic Properties of Real Numbers. .35. .. .. .3535392932xiiixivContents2.1.3The Completeness Axiom and the Existence of a LeastUpper (or Greatest Lower) Bound of a Set of Numbers . .2.2 The Most Important Classes of Real Numbersand Computational Aspects of Operations with Real Numbers . .2.2.1 The Natural Numbers and the Principle of MathematicalInduction .
. . . . . . . . . . . . . . . . . . . . . . . . .2.2.2 Rational and Irrational Numbers . . . . . . . . . . . . .2.2.3 The Principle of Archimedes . . . . . . . . . . . . . . .2.2.4 The Geometric Interpretation of the Set of Real Numbersand Computational Aspects of Operations with RealNumbers . . . . .
. . . . . . . . . . . . . . . . . . . . .2.2.5 Problems and Exercises . . . . . . . . . . . . . . . . . .2.3 Basic Lemmas Connected with the Completeness of the RealNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.1 The Nested Interval Lemma (Cauchy–Cantor Principle) .2.3.2 The Finite Covering Lemma (Borel–Lebesgue Principle,or Heine–Borel Theorem) . .
. . . . . . . . . . . . . . .2.3.3 The Limit Point Lemma (Bolzano–Weierstrass Principle)2.3.4 Problems and Exercises . . . . . . . . . . . . . . . . . .2.4 Countable and Uncountable Sets . . . . . . . . . . . . . . . . .2.4.1 Countable Sets . . . .
. . . . . . . . . . . . . . . . . . .2.4.2 The Cardinality of the Continuum . . . . . . . . . . . . .2.4.3 Problems and Exercises . . . . . . . . . . . . . . . . . .34Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1 The Limit of a Sequence . . . . . . . . . . . . . . . . . . . .
. .3.1.1 Definitions and Examples . . . . . . . . . . . . . . . . .3.1.2 Properties of the Limit of a Sequence . . . . . . . . . . .3.1.3 Questions Involving the Existence of the Limitof a Sequence . . . . . . . . . . . . . . . . . . . . . . .3.1.4 Elementary Facts About Series . . . . . . . .
. . . . . .3.1.5 Problems and Exercises . . . . . . . . . . . . . . . . . .3.2 The Limit of a Function . . . . . . . . . . . . . . . . . . . . . .3.2.1 Definitions and Examples . . . . . . . . . . . . . . . . .3.2.2 Properties of the Limit of a Function . . . . . . . . . . .3.2.3 The General Definition of the Limit of a Function (Limitover a Base) . .
. . . . . . . . . . . . . . . . . . . . . .3.2.4 Existence of the Limit of a Function . . . . . . . . . . .3.2.5 Problems and Exercises . . . . . . . . . . . . . . . . . .Continuous Functions . . . . . . . . . . . . .4.1 Basic Definitions and Examples . . . . . .4.1.1 Continuity of a Function at a Point4.1.2 Points of Discontinuity . .
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