An Introduction to Analysis

The book contains a rigorous exposition of calculus of a single real variable. It covers the standard topics of an introductory analysis course, namely, functions, continuity, differentiability, sequences and series of numbers, sequences and series of functions, and integration. A direct treatment of the Lebesgue integral, based solely on the concept of absolutely convergent series, is presented, which is a unique feature of a textbook at this level. The standard material is complemented by topics usually not found in comparable textbooks, for example, elementary functions are rigorously defined and their properties are carefully derived and an introduction to Fourier series is presented as an example of application of the Lebesgue integral.

The text is for a post-calculus course for students majoring in mathematics or mathematics education. It will provide students with a solid background for further studies in analysis, deepen their understanding of calculus, and provide sound training in rigorous mathematical proof.

Contents:

• Preface
• Real Numbers
• Limits of Functions
• Continuous Functions
• Derivatives
• Sequences and Series
• Sequences and Series of Functions
• Elementary Functions
• Limits Revisited
• Antiderivatives
• The Lebesgue Integral
• Miscellaneous Topics
• Construction of Real Numbers

Readership: Undergraduate students taking the introductory real analysis course.
Key Features:

• The textbook presents a rigoruous development of the Lebesgue integral that is based solely on the concept of absolutely convergent series, completely avoiding the introduction of measure theory. This is a unique feature of a textbook at this level
• A novel construction of the real numbers from natural numbers is presented. Instead of the standard Peano's axioms, a new, simpler approach is used: Axioms for the natural numbers are formulated in terms of inequalities
• The book has a large number of original exercises of the type not usually seen in introductory analysis textbook