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 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!

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 got this book for my son. He's in 11th grade. He found the book to be very helpful. It was much simpler than the MIT Real Analysis Lectures on YouTube and his other book "Real and Complex Analysis" by Shilov. He said that it is a good book to get him started and it's providing him with a solid foundation for the hard stuff. There were some dents in the pages and it looked like it had been mishandled but everything else is perfect. I usually expect to order something new and get it in new condition though.

Ive been trying to self study real analysis. Here’s a list of some analysis books I've either read or skimmed through: - Understanding Analysis by Abbott - fundamentals of mathematical analysis Haggarty - analysis with an introduction to proof by Lay - introduction to real analysis by Silva - introduction to analysis by Mattuck - many, many more that i could get my hands on This is the best one i could find. The author actually explains the motivation behind the proofs, and shows you how to *think* about them so you can derive the theorem and similar results on your own. This is in stark contrast to the presentation in many other books where a pretentious “polished” proof is shown with absolutely no hint as to *how* it was actually derived in the first place. Not so with this book. The author shows how the derivation is actually done by a human and not a computer. There are also plenty of illustrations in the book to provide an intuitive understanding of the proofs before giving the formal derivation. Each chapter comes with an introductory section providing the motivation for the topics about to be studied, which gets you curious about what you’re about to learn. This is the first math book I’ve read that I actually want to *read* for its own sake, not just use as something to “study” from. The only shortcoming is that there are no solutions to the exercises in the book, which often include crucial concepts (none that are needed for later chapters though; you can easily read the entire book and work on the examples without doing the exercises). The author does provide hints and partial solutions to some of the exercises on his website but i wish they were more comprehensive for those looking to self study. I also wish there was a section on metric spaces but really, i cant complain. This is such a well written book; it deserves nothing less than 5 stars. I haven’t listed the many books i tried to work through to understand analysis, this one just clicked for me very quickly. I even prefer it over Abbott’s as a first book on mathematical analysis, and I’ll be buying his other book on proof writing as well :)

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.

Book arrived within just a couple of days of ordering. A difficult subject and well explained. It's a simplified approach and the author's more conversational style is more user friendly than the standard theorem-proof-theorem-proof approach of the traditional authors who follow Euclid's writing style, much as a classical pianist learns to play scales for practice and fingering techniques for more advanced keyboard work. One point that might be noteworthy is that the classroom courses frequently pick a different consistent set of axioms than the textbook uses. The reason for this is to prevent copying of proofs by students from one semester to the next or from other textbook authors they may use for additional references. In many courses, you will be expected to figure this out on your own and you should understand that the textbooks used are for examples of how a proof should be written. But you will have to come up with your own proof at each step along the way. This is painful and tedious but a necessary step in becoming a real mathematician. Also, one of the mistakes that students often make is to not base their subsequent proofs on what they have already proved. You only need cite the earlier proof(s) you are starting from and build on them wherever possible. Just use care in doing so and be sure you provide a reference to each previous proof you use. For exams, you may need to present the entire thing in gory detail depending on what your instructor expects. Lastly, the attrition rate in a course of this sort is frightening. You cannot let it get to you if you expect to pass. The learning curve is usually very steep and you will often find you are humiliated and humbled as you struggle through the class, You are not alone in this feeling. Pick yourself up and dust yourself off when you fall on your face. Cry a little if you must and then get back in the game!

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.

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.