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I am looking for a book on (one variable) real analysis that includes, simultaneously:

  • A treatment of the abstract and theoretical aspects of real analysis;
  • A treatment of the more mechanical and computational aspects of calculus (such as techniques of antidifferentiation, for instance).

The reason I'm finding this hard to find is that usually undergraduates take a course in calculus where they learn mostly the mechanics and then take a real analysis course where the emphasis is on theory. I'm looking for a book that integrates these two aspects. Ideally the logical prerequisites should be (besides some mathematical maturity) the mathematics one learns in high school (algebra, trigonometry, ...).

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Try Spivak's Calculus. –  William Jul 30 '12 at 3:56
I'd also recommend Spivak, but further recommend that you have a purely "mechanics of calculus" text in mind if you're just coming out of high school math. Spivak's Calculus is an excellent book, but most would find it a little cruel for a first go at the subject. –  Robert Mastragostino Jul 30 '12 at 4:05
I know Spivak's book and I agree that it is an excellent text, but it's not quite what I have in mind. I was thinking more about a book that covers approximately the same material as Rudin's "Principles of Mathematical Analysis", while also including the mechanical aspects of calculus (for instance, Rudin doesn't include techniques of antidifferentiation because he assumes that the reader has obviously already learned it in a calculus course). –  Filipe G. Jul 30 '12 at 4:35
Why, exactly, are you looking for such a beast? –  dls Jul 30 '12 at 4:44
@dls: In my country, there is usually no such thing as a "calculus sequence" followed by a course in real analysis. Math majors take three to four one-semester courses in real analysis where the theoretical and mechanical aspects are treated together. For instance, in my first couple of real analysis course I learned stuff like the topology of the real line and Lebesgue's criterion for Riemann integrability while also learning how to compute integrals using the techniques usually found in calculus courses. Purely mechanical calculus courses are usually taken by non-mathematics majors only. –  Filipe G. Jul 30 '12 at 4:53

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