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Introduction to Analysis (Dover Books on Mathematics)

Maxwell Rosenlicht · 3 HN comments
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Amazon Summary
This well-written text provides excellent instruction in basic real analysis, giving a solid foundation for direct entry into advanced work in such fields as complex analysis, differential equations, integration theory, and general topology. The nominal prerequisite is a year of calculus, but actually nothing is assumed other than the axioms of the real number system. Because of its clarity, simplicity of exposition, and stress on easier examples, this material is accessible to a wide range of students, of both mathematics and other fields. Chapter headings include notions from set theory, the real number system, metric spaces, continuous functions, differentiation, Riemann integration, interchange of limit operations, the method of successive approximations, partial differentiation, and multiple integrals. Following some introductory material on very basic set theory and the deduction of the most important properties of the real number system from its axioms, Professor Rosenlicht gets to the heart of the book: a rigorous and carefully presented discussion of metric spaces and continuous functions, including such topics as open and closed sets, limits and continuity, and convergent sequence of points and of functions. Subsequent chapters cover smoothly and efficiently the relevant aspects of elementary calculus together with several somewhat more advanced subjects, such as multivariable calculus and existence theorems. The exercises include both easy problems and more difficult ones, interesting examples and counter examples, and a number of more advanced results. Introduction to Analysis lends itself to a one- or two-quarter or one-semester course at the undergraduate level. It grew out of a course given at Berkeley since 1960. Refinement through extensive classroom use and the author’s pedagogical experience and expertise make it an unusually accessible introductory text.
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Digression: Terry Tao wrote a book on analysis? That's awesome!


I wouldn't recommend Spivak for self-study and a first exposure to analysis. It's known to be notoriously difficult even for good students.

I learned from this Dover book[1]; I think it's pretty good. From there you might move up to Baby Rudin[2], but it might have a lot of typos, or big gaps in the exposition that are taken to be obvious but require several steps to fill in, since that was certainly the case for Papa Rudin[3].

[1]: [2]: [3]:

I highly recommend this book for Real Analysis

We used it at the University of Texas at Austin for the first semester in Real Analysis. I found it very clear and easy to follow.

I used this for my undergrad analysis class and then relied on it as a foundation for reading Rudin/Royden in grad school. Killed qualifying exams. Use it. Sure, it's all Riemann integration but you will not suffer until you get to measure-theoretic integration later.
Why are these books from Dover Publications so cheap?
Because the copyright has expired, or the copyright owner has put the work in the public domain due to bankruptcy, or similar.
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