Real Analysis: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series) 2nd Edition

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 just finished reading a most wonderful book on Real Analysis by Jay Cummings. Asunusual as it may seem, I normally scan over a book before I begin reading it. In thiscase I actually read the Appendices first, before I began reading the book. The Appendiceswere fantastic and I had to keep reading them because they were so good. As an undergraduateI learned how to construct the real numbers by starting with nothing but the empty set. Jaydoes 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 RealAnalysis. You will know Jay is an expert when you finish Appendix B. The book's cover showsthe graph of what is called Thomae's function (I had not seen this before in any of thebooks I have on Real Analysis). Later he proves this function is integrable. Strangelyenough, I am probably one of the few people to take a full year course in Topology beforeI took my first course in Real Analysis. I found Topology fascinating, but you aren'tsupposed to take Topology before you take Real Analysis. How ironic then that in Jay's bookChapter 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 outstandingproblems that will really help you learn the subject He also includes many open questions that willkeep you entertained. I only wish I could have had this book when I took Real Analysis. It wouldhave made my life much easier!

I really enjoy the author’s writing style. It’s full of personality and stands out from the typical math textbook. You can genuinely sense the author’s passion for mathematics throughout the book. Unlike many Real Analysis texts that strictly follow a dry Theorem-Proof-Theorem-Proof format, this one brings a refreshing, more engaging approach. That said, if you prefer a more traditional and formal presentation, this book might feel a bit unconventional. For self-study, I strongly recommend having taken a course in Discrete Mathematics. If it’s been a while, brushing up on proof techniques will be helpful. Fortunately, the author includes helpful hints along the way, so be sure to take advantage of those.

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.