Reviewed by Sonmez Sahutoglu, Associate Professor, University of Toledo on 8/22/16

Comprehensiveness
rating: 4 see less

This text covers all the standard material for a senior level undergraduate (or master level) real analysis class. It start with basics set theory and the real numbers. Then it develops supremum and infimum of bounded sets; limit, limsup, liminf for sequences; series of numbers and functions; continuity and uniform continuity of functions, derivative of functions, Riemann integral and improper integrals, basics of metric spaces including connectedness, compactness set and continuity of functions between metric spaces. Finally it ends with a proof of fixed point theorem. The text covers all the main theorems (such as mean value theorem, intermediate value theorem, Heine-Borel theorem, Bolzano-Weierstrass theorem, Dini’s theorem) one would expect to be covered in this area. It includes a reasonable number of problems and examples. The text provides an effective index at the end.

Accuracy
rating: 5

I did not see anything inaccurate.

Relevance/Longevity
rating: 5

I don't expect the text to be obsolete any time soon as this is an advanced math textbook.

Clarity
rating: 5

The material is presented in a clear fashion and should be very readable by students.

Consistency
rating: 5

The style of the book is very consistent throughout the book.

Modularity
rating: 4

It is as modular as one can expect of a book on this subject. Since everything numbered consistently one can easily find the results needed from the previous chapters.

Organization/Structure/Flow
rating: 5

The material is presented in a logical order. Definitions, theorems, examples, exercises, etc are all numbered in a consistent manner. The book is written in a little bit informal language but that is not a shortcoming.

Interface
rating: 4

In general, the interface of this book is very typical of an advanced math textbook. All the statements are numbered and hyperlinked making navigation very easy.

Grammatical Errors
rating: 5

Other then few typos I did not see any grammatical errors.

Cultural Relevance
rating: 5

This is not an issue for graduate level mathematics books and this book is no exception.

Comments

This book has been approved by the American Institute of Mathematics Open Textbook Initiative. I have used it twice in my classes and have been very happy about it. The author maintains an errata on his website and has been updating the text regularly. I suggest using the latest edition that can be obtained from the authors website.

Reviewed by William Newton, Research Scientist, Colorado State University on 1/8/16

Comprehensiveness
rating: 5 see less

The textbook covers everything I would want to cover in a course at this level and some more. A teacher can use this book as the sole text of an introductory analysis class and skip the last two chapters for a slower class.

Accuracy
rating: 4

I found a few minor typographical errors but no serious problems.

Relevance/Longevity
rating: 5

This is exactly the material that should be covered in an introductory analysis class and it will remain relevant up to date for a long time.

Clarity
rating: 5

The material is presented in a clear, concise manner. Every term is adequately defined.

Consistency
rating: 5

The author does a great job of remaining consistent with terminology throughout the book.

Modularity
rating: 5

It is difficult to write a highly modular textbook with this material since many results depend on earlier ones. However, this book does as good of a job as could be expected by breaking down the material into reasonable sections and maintaining a consistent numbering scheme throughout the book. When earlier results need to be referenced, they are easy to find.

Organization/Structure/Flow
rating: 5

I am glad that the book opens with set theory. The lack of such material can be a shortcoming of some analysis texts. The chapters are presented in a logical order, with all of the material building on previous results.

Interface
rating: 5

I had no trouble reading this book or finding results. I appreciate the use of a consistent numbering scheme for results throughout the textbook.

Grammatical Errors
rating: 4

I found a few typographical errors, but otherwise everything was fine.

Cultural Relevance
rating: 5

This material should be appropriate in any classroom where this subject is being taught. I saw nothing that seemed culturally insensitive.

Comments

This is a great textbook for introductory analysis, and I expect that I will use it the next time I teach the subject.