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This is a text for a course in introductory real analysis for junior or senior mathematics majors and science students with a serious interest in mathematics. Prospective educators or mathematically gifted high school students can also benefit from the mathematical maturity that can be gained from an introductory real analysis course. The book is designed to fill the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics.
The standard elementary calculus sequence is the only specific prerequisite for Chapters 1–5, which deal with real-valued functions. (However, other analysis oriented courses, such as elementary differential equation, also provide useful preparatory experience.) Chapters 6 and 7 require a working knowledge of determinants, matrices and linear transformations, typically available from a first course in linear algebra. Chapter 8 is accessible after completion of Chapters 1–5.
Includes 295 completely worked out examples to illustrate and clarify all major theorems and definitions.
The Real Numbers.
The Real Number System.
Mathematical Induction.
The Real Line.
Differential Calculus of Functions of One Variable.
Functions and Limits.
Continuity.
Differentiable Functions of One Variable.
L’Hospital’s Rule.
Taylor’s Theorem.
Integral Calculus of Functions of One Variable.
Definition of the Integral.
Existence of the Integral.
Properties of the Integral.
Improper Integrals.
A More Advanced Look at the Existence of the Proper Riemann Integral.
Infinite Sequences and Series.
Sequences of Real Numbers.
Earlier Topics Revisited With Sequences.
Infinite Series of Constants.
Sequences and Series of Functions.
Power Series.
Real-Valued Functions of Several Variables.
Structure of Rn.
Continuous Real-Valued Function of n Variables.
Partial Derivatives and the Differential.
The Chain Rule and Taylor’s Theorem.
Vector-Valued Functions of Several Variables.
Linear Transformations and Matrices.
Continuity and Differentiability of Transformations.
The Inverse Function Theorem.
The Implicit Function Theorem.
Integrals of Functions of Several Variables.
Definition and Existence of the Multiple Integral.
Iterated Integrals and Multiple Integrals.
Change of Variables in Multiple Integrals.
Metric Spaces.
Introduction to Metric Spaces.
Compact Sets in a Metric Space.
Continuous Functions on Metric Spaces.
Answers to Selected Exercises.
Functions Defined by Improper Integrals:
Foreword.
Preparation.
Uniform convergence of improper integrals.
Absolutely Uniformly Convergent Improper Integrals.
Dirich let's Tests.
Consequences of uniform convergence.
Applications to Laplace transforms.
Exercises.
Answers to selected exercises