A Course in Real Analysis

“A Course in Real Analysis” by an unnamed author provides a comprehensive study of the foundations of differential and integral calculus at the advanced undergraduate level. The book has been thoroughly tested in the author’s course on real analysis at The George Washington University. It is structured in three parts to cover topics ranging from single-variable calculus to multivariable functions.

The first part of the text delves into the calculus of functions of one variable. It covers traditional subjects such as sequences, continuity, differentiability, Riemann integrability, numerical series, and the convergence of functions and sequences. Additionally, there are optional sections on advanced topics like Stirling’s formula, bounded variation functions, Riemann–Stieltjes integration, and more.

The second part shifts focus to functions of multiple variables and introduces topological concepts essential for analyzing multivariable functions. It discusses differentiability, integrability, and also explores the theory of differential forms on surfaces in ℝn.

The book concludes with appendices on set theory and linear algebra, along with solutions to select exercises. Instructors have the option of utilizing a comprehensive solutions manual that provides answers to all exercises in the text.

With detailed proofs, numerous examples, and a wide range of exercises, this textbook offers a thorough exploration of real analysis. It guides students from single-variable calculus to multivariable functions in a logical progression, preparing them for more advanced research-oriented courses.

**FAQs**

**1. What topics are covered in “A Course in Real Analysis”?**
The book covers calculus of functions of one variable, functions of multiple variables, topological concepts, and differential forms among other topics.

**2. Is there a solutions manual available for instructors?**
Yes, a full solutions manual providing answers to all exercises in the book is available for instructors.

**3. Are there exercises included in the book for practice?**
Yes, the textbook includes numerous exercises to help students practice and reinforce their understanding of the material.

**4. How is the book structured in terms of content?**
The book is divided into three parts covering single-variable calculus, multivariable functions, and additional appendices on set theory and linear algebra.

**Conclusion**

“A Course in Real Analysis” is a valuable resource for students and instructors looking to deepen their understanding of calculus at the undergraduate level. With its comprehensive coverage of both single-variable and multivariable functions, along with clear explanations and ample exercises, the book serves as a solid foundation for further studies in mathematics.