The book begins with a preliminary chapter on sets, functions, and real numbers. After briefly providing basic set theory, we introduce the completeness property of real numbers and its several consequences. The main body consists of two parts. Part I is devoted to developing the theory of functions of one variable. Functions of several variables which are more difficult and technical are studied in Part II. Part I is self-contained. All the theorems in Part I are deduced from the completeness property of real numbers. But some results in Part II are proved by using standard results from linear algebra such as elementary matrices and determinants. Therefore, readers of Part II should have some knowledge about elementary Linear Algebra. It is also highly recommended to be familiar with basic set theory in the preliminary chapter since set-theoretical arguments are inevitably used to establish many important theorems for functions of several variables. There are two kinds of problems in this book: exercises inside each section and problems at the end of each chapter. Section exercises are relatively easy and provided for self-study. Making efforts to solve exercises by trial and error, readers not only better understand new concepts and theorems in sections but also have chances to generalize them to some extent. Chapter problems are placed at the end of each chapter because materials from more than two sections are often necessary to solve them successfully. Some problems allow readers to obtain interesting new results by applying the concepts and results in the chapter. However, chapter problems are more challenging than section exercises. Problems marked by an * are provided with short hints at the end of the book.

저자는 고등 미적분학을 배우기에 앞서 필요한, 집합과 함수, 실수 등을 다루며 책을 시작합니다. 기초 집합론이 간략히 제시된 후 실수의 완비성과 그에 따른 여러 결과가 소개됩니다. 본문은 두 편으로 구성됩니다. 제1편에서는 일변수 함수 이론을 전개하는 데 중점을 두었습니다. 더 어렵고 전문적인, 다변수 함수는 제2편에서 배웁니다. 제1편은 자립적(self-contained)입니다. 제1편의 모든 정리는 실수의 완비성(completeness property)에서 연역됩니다. 그러나 제2편의 몇몇 결과는 행렬 및 행렬식과 같은, 선형 대수학에서 도출된 표준 결과로써 증명됩니다. 그러므로 제2편을 읽는 이는 일정 수준의 기초 선형 대수학 지식을 갖춰야 합니다. 또한, 독자들은 집합론의 기본적인 부분에 익숙해질 필요가 있습니다. 왜냐하면, 집합론적 논의는 다변수 함수의 중요한 정리들을 세우는 데 반드시 쓰이기 때문입니다. 본서에는 두 가지 종류의 문제가 있습니다. 각 섹션에 포함된 Exercises와 챕터 마지막에 배치된 Problems가 그것입니다. 섹션별 Exercises는 상대적으로 쉬운 문제로 구성되어 있으며 자습용으로 제공됩니다. 시행착오를 거치며 문제를 해결함으로써 독자들은 해당 섹션의 개념과 정리를 보다 효과적으로 이해할 수 있을 뿐만 아니라 그러한 개념, 정리들을 다소간 일반화할 기회를 얻게 됩니다. Problems는 각 챕터의 마지막에 실렸는데, 그 이유는 문제를 성공적으로 풀기 위해선 종종 두 섹션 이상의 내용이 요구되기 때문입니다. Problems의 문제는 독자가 그 챕터의 개념과 결과를 적용하여 흥미롭고 새로운 결과를 얻을 수 있게 합니다. 하지만 Problems의 문제들은 Exercises의 문제보다 도전적입니다. *표시가 있는 문제에 대한 짧은 힌트가 책 뒷부분에 수록되어 있습니다.

Preface

Preface to the Second Edition

0 Preliminaries

0.1 Sets and Functions

0.2 Mathematical Induction

0.3 Countable and Uncountable Sets

0.4 Completeness of Real Numbers

0.5 Consequences of the Completeness

0.6 Problems

I Functions of One Variable

1 Sequences and Series

1.1 Limits of Sequences

1.2 Limit Theorems

1.3 The Bolzano-Weierstrass Theorem

1.4 Cauchy Sequences

1.5 Monotone Sequences

1.6 Limits Superior and Inferior

1.7 Series of Real Numbers

1.8 Convergence Tests for Series

1.9 Problems

2 Limits and Continuity

2.1 Limits of Functions

2.2 Continuous Functions

2.3 Uniformly Continuous Functions

2.4 Monotone Functions

2.5 Problems

3 Differentiation

3.1 Derivatives

3.2 Differentiation Rules

3.3 Exponential and Logarithmic Functions

3.4 Mean Value Theorem

3.5 L’Hospital’s Rule

3.6 Taylor’s Theorem

3.7 Problems

4 Integration

4.1 Riemann Integrals

4.2 Properties of Integrals

4.3 Further Properties of Integrals

4.4 Fundamental Theorems of Calculus

4.5 Improper Integrals

4.6 Problems

5 Sequences and Series of Functions

5.1 Double Series

5.2 Pointwise and Uniform Convergence

5.3 Tests for Uniform Convergence

5.4 Consequences of Uniform Convergence

5.5 Power Series

5.6 Taylor Series

5.7 Problems

II Functions of Several Variables

6 Euclidean Spaces

6.1 The Euclidean Space Rn

6.2 The Bolzano-Weierstrass Theorem in Rn

6.3 Open and Closed Sets

6.4 Compact Sets

6.5 Connected Sets

6.6 Problems

7 Continuity of Multivariable Functions

7.1 Limit and Continuity in Rn

7.2 Properties of Continuous Functions

7.3 Contractions

7.4 Linear Functions

7.5 The Weierstrass Approximation Theorem

7.6 Problems

8 Differentiation of Multivariable Functions

8.1 Partial Derivatives

8.2 Differentiability

8.3 Chain Rule

8.4 Mean Value Theorem in Rn

8.5 Inverse Function Theorem

8.6 Implicit Function Theorem

8.7 Optimization

8.8 Problems

9 Integration of Multivariable Functions

9.1 Integrals on Boxes

9.2 Integrals on General Sets

9.3 Sets of Volume Zero and Integrable Functions

9.4 Iterated Integrals

9.5 Change of Variables: Preliminary Lemmas

9.6 Change of Variables

9.7 Evaluation of Some Integrals

9.8 Problems

Bibliography

Hints for Problems

Index 

Hyunseok Kim

Department of Mathematics, Sogang University (Ph. D., POSTECH)

Young-Ran Lee

Department of Mathematics, Sogang University (Ph. D., University of Alabama at Birmingham)

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