Table of Contents
Chapter 1: Tools for Analysis
1.1 Basic Concepts of Set Theory
1.3 The Natural Numbers and Mathematical Induction
1.4 Order Field Axioms
1.5 The Completeness Axiom for the Real Numbers
1.6 Applications of the Completeness Axiom
Chapter 2: Sequences
2.2 Limit Theorems
2.3 Monotone Sequences
2.4 The Bolzano-Weierstrass Theorem
2.5 Limit Superior and Limit Inferior
2.6 Open Sets, Closed Sets, and Limit Points
Chapter 3: Limits and Continuity
3.1 Limits of Functions
3.2 Limit Theorems
3.4 Properties of Continuous Functions
3.5 Uniform Continuity
3.6 Lower Semicontinuity and Upper Semicontinuity
Chapter 4: Differentiation
4.1 Definition and Basic Properties of the Derivative
4.2 The Mean Value of Theorem
4.3 Some Applications of the Mean Value Theorem
4.4 L’Hospital’s Rule
4.5 Taylor’s Theorem
4.6 Convex Functions and Derivatives
4.7 Nondifferentiable Convex Functions and Subdifferentials
Chapter 5: Solutions and Hints for Selected Exercises
About the Book
Our goal with this textbook is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs.
The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. The lecture notes also contain many well-selected exercises of various levels. Although these topics are written in a more abstract way compared with those available in some textbooks, teachers can choose to simplify them depending on the background of the students. For instance, rather than introducing the topology of the real line to students, related topological concepts can be replaced by more familiar concepts such as open and closed intervals. Some other topics such as lower and upper semicontinuity, differentiation of convex functions, and generalized differentiation of non-differentiable convex functions can be used as optional mathematical projects. In this way, the lecture notes are suitable for teaching students of different backgrounds.
The second edition includes a number of improvements based on recommendations from students and colleagues and on our own experience teaching the course over the last several years.
In this edition we streamlined the narrative in several sections, added more proofs, many examples worked out in detail, and numerous new exercises. In all we added over 50 examples in the main text and 100 exercises (counting parts).
About the Contributors
Beatriz Lafferriere, Assistant Professor, Fariborz Maseeh Department of Mathematics and Statistics, Assistant Chair for Undergraduate Program, Director of Undergraduate Advising, Portland State University. PhD Rutgers University.
Gerardo Lafferriere, Professor, Fariborz Maseeh Department of Mathematics and Statistics, Portland State University. PhD Rutgers University. Area of Specialty: Mathematical Control Theory, Hybrid Systems, Mathematical Biology, Robotics
Mau Nam Nguyen, Associate Professor, Fariborz Maseeh Department of Mathematics and Statistics, Portland State University. Ph.D. 2007 Wayne State University. Area of Specialty: Variational & Convex Analysis, Mathematical Optimization, Non-Linear & Functional Analysis.