Real Analysis: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series)

As a student at one of the University of California schools taking Real Analysis, this book is perfect for both following along with the class and self-studying. I have been a mostly self-taught student, reading books prior to my classes, and found this book to be very engaging. It's an enjoyable read, providing insightful quips to keep your interest piqued. Context is thoroughly provided before entering a topic - whether it be historical or math relevant. Footnotes contain interesting comments along with additional commentary on harder topics; sometimes even jokes. Honestly, most books fail to connect to the reader - like they're some robot, but when I read this it's as if I'm talking to Jay Cummings himself. It's a human-to-human read. So far, Real Analysis tends to be hard because your intuition fails you at times. The book does its best to supplement the occasional topics that deceive your intuition. Most of the reviews seem to complain about not having enough examples, but in reality, there are plenty (along with ALOT of additional notes in footnotes!). There are also solutions posted to the exercises online (on the author's site iirc). There have been times when I couldn't understand something specific and had to seek other material on YouTube (and this is normal). Some topics click to others and some don't. Ultimately, you'll find yourself understanding 90% of the book alone. That's just how Real Analysis is and there are some parts of math that will be harder to understand and require extra care. For example, when the book covers convergent sequences there is a great emphasis on understanding the definitions and even gives you multiple "comments". Each comment provides a different perspective than the one before and ultimately gives you the best opportunity to learn. (I've attached an image of part of the convergent sequences). Something unique that the book does is it gives you a page of contents for every proposition, lemma, and definition given. Truly a convenience. Ultimately, this book rules. If you're a like-minded student, this is perfect for you. Provides amazing intuition and historical context which helps you understand the purpose of the math you are learning. The book is also easily read and funny. I rarely write reviews but I couldn't pass this book up. Good luck with Real Analysis. Also, if it means anything, The Math Sourcerer on Youtube reviewed this book and practically gave it a 10/10. So if my review doesn't convince fellow students, check out his review. Way more in-depth than mine probably.

I'm a mathematics and computer science undergraduate student and have found this book (and the mathematical proofs book) incredibly helpful. The content is rigorous while also interspersed with humor and interesting footnotes that help alleviate my "math anxiety." The scratch work, proofs, and layout are clear and approachable. If you're taking real analysis you should definitely add this book to your collection.

I just finished reading a most wonderful book on Real Analysis by Jay Cummings. As unusual as it may seem, I normally scan over a book before I begin reading it. In this case I actually read the Appendices first, before I began reading the book. The Appendices were fantastic and I had to keep reading them because they were so good. As an undergraduate I learned how to construct the real numbers by starting with nothing but the empty set. Jay does the same thing in Appendix A, but he cleverly avoids boring you with all the details. His summary is succinct and to the point and it is all outlined in just a few pages. His Appendix B contains classic pathological examples which motivate most of the subject of Real Analysis. You will know Jay is an expert when you finish Appendix B. The book's cover shows the graph of what is called Thomae's function (I had not seen this before in any of the books I have on Real Analysis). Later he proves this function is integrable. Strangely enough, I am probably one of the few people to take a full year course in Topology before I took my first course in Real Analysis. I found Topology fascinating, but you aren't supposed to take Topology before you take Real Analysis. How ironic then that in Jay's book Chapter 5 is a brief intro to Topology and this is before Chapter 6 which discusses Continuity. Little wonder that I felt right at home with this book. The author also very carefully introduces the concept of integrability and touches on measure theory. You immediately learn why the author is such an expert. The rest of the book is full of outstanding problems that will really help you learn the subject He also includes many open questions that will keep you entertained. I only wish I could have had this book when I took Real Analysis. It would have made my life much easier!

This is a great gentle introduction to real analysis. The range of topics neither deep nor detailed, but it's an entertaining and easy-to-read overview of what analysis is all about. You won't learn the differences between Darboux, Reimann-Stiltjes, and Lebesque integrals, but you will learn what is going on behind the scenes of the introductory calculus courses you took.

Fantastic textbook. Well written and easy to follow along. Very likely you will still need some supplementation with ChatGPT and other resources (I did) but this is a good anchor for learning Real Analysis if you're serious about improving your math skills.

good way of presenting the matter

It costs a little higher vis a vis Indian market ,but it is worth every penny spent , because it teaches you , in a cheerful mood. Author not only presents the contents within a comprehensive single thread , he also breaks ur boredom , if any, with his witty remarks and humour ! Great book, strongly recommended. It is actually an introductory real analysis book, but deserves appreciation. Page and printing worth appreciation as well.

Very well written.

The author introduces all central proofs required for a one-semester long module on real analysis in an easily understandable and comprehensive approach. I will facilitate mastering the challenging, abstract syllabus of any undergraduate maths curriculum. However, every year-long modules will cover more topics and at greater depth, requiring an additional good textbook like Elementary Real Analysis by Bruckner, Thompson, Thompson. Nonetheless, the value of the book is at least 10 times as high as its very fair price, and it ought to be purchased as a treatise that covers the main proofs in all steps and details - way better than any video-course on YouTube.

Questo testo è il link ideale fra i corsi di Calculus e Real Analysis così come vengono chiamati in UK/US. In Italia siamo meno abituati a questa distinzione, perchè almeno i nostri vecchi corsi di Analisi assumevano l'una e l'altra veste. In ogni caso, sia nello studio da autodidatta che di preparazione preliminare a corsi più avanzati, questo ottimo libro accompagna il lettore nel viaggio alla scoperta delle strategie di dimostrazione e del linguaggio tipico dell'Analisi Matematica. L'inglese utilizzato è davvero di facile comprensione e l'esposizione è informale quando opportuno, al tempo stesso è rigorosa nella presentazione delle definizioni, teoremi e dimostrazioni finali. Per alcune dimostrazioni si adotta una strategia graduale, prima una bozza informale e alla fine la "vera" dimostrazione, inutile dire come questo abbia un gran valore pedagogico, cosa di cui i testi più comuni devono fare necessariamente a meno. Il contenuto tocca gli argomenti tipici di un primo corso di Analisi: numeri reali, successioni e serie numeriche, limiti, continuità, differenziazione, integrazione, successioni e serie di funzioni. Consiglio di utilizzarlo come accompagnamento ai testi tradizionali sugli stessi argomenti.