This book is meant for those who have studied one-variable calculus (and maybe higher-level courses as well), generally skipping the proofs in favor of learning the techniques and solving problems. Now they are interested in learning to read proofs, and to find and write up their own: perhaps because they will need this for the next steps in their chosen field, or for intellectual satisfaction, or just out of curiosity.
There are two paths to this. Some books start with a great leap forward, giving the definitions in n-space. This requires an excursion into point-set topology, whose proofs are unlike those of the usual calculus courses and are a roadblock to many.
The path chosen by this book is to start like calculus does, in 1-space (i.e., on the line) and focus on the basic definitions and ideas of one-variable calculus: limits, continuity, derivatives, Riemann integrals, and a few more advanced topics. It's done rigorously, but also in as familiar a way as possible. So from the start it will use as a source of examples what you know (with occasional reminders): K-12 mathematics and basic one-variable calculus, including the log, exp, and trig functions. This takes up about two-thirds of the book, and might be as far as you wish to go. It sounds like just repeating calculus, but students say that it feels very different and is not all that easy.
The rest of the book gets into ideas from advanced calculus used in lower-level courses without proof: uniform convergence, differentiating infinite series term-by-term and integrals containing a parameter (the Laplace transform, for instance). For the latter, it's finally time to learn about point-set topology in the plane (2-space, but n-space is no harder). There's also for the curious or needy an optional chapter with the most important facts about point-sets of measure zero on the line and a more powerful integral, the Lebesgue integral. Two appendices respectively provide needed and optional background in elementary logic, and four more give interesting applications and extensions of the book's theory.
For more details, click on "Look Inside" to see the Table of Contents.
Some generally helpful features:
--Leisurely exposition, with serious comments about proofs, other possible arguments, writing advice; some semi-serious comments too;
--Attention paid to layout and typography, both for greater readability, and to give readers models they can imitate;
--Questions after most sections of a chapter to firm up what you just read, with Answers of various sorts at the end of the chapter: single words, hints, complete statements, formal proofs.
Mathematically helpful features:
--The language of limits is simplified by suppressing the N and the delta when their explicit value is not needed in the argument, replacing them with standard applied math symbols meaning "for n large" and "for x sufficiently close to a". These are introduced carefully and rigorously; some caution is needed, which is described at the end of the Preface (click "Look Inside").
--The book tries to go back to the roots of real analysis by emphasizing estimation and approximation, which use inequalities rather than the equalities of calculus, but have a similar look, so that many proofs are calculation-like "derivations" that seem familiar. But inequalities are often mishandled and warnings are given.
For examples of these features and writing style, go to the author's home page, link to "book", then link to "sample pages" from the first three chapters.
The book was developed at MIT, mostly for students not in mathematics having trouble with the usual real-analysis course. It has been used at large state universities and at small colleges, as well as for independent study. Students evaluate it as readable and helpful. The new printing, by CreateSpace and at a reduced price, is the eighth, incorporating all known significant corrections and a new Appendix 6.
