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I'm looking for a good introductory text to analysis, or, more specifically, a text that puts calculus on a much more rigorous ground. I've just finished a year of calculus at my local university, and I feel a bit cheated by the course I took. In addition to having a poor teacher who focused on rote memorization, our text was abysmal. I ended up resorting to self study, which was a blessing in disguise because it led me to discover a love of logic, set theory, abstract algebra, and topology.

I'm interested in becoming a math major when I graduate hs, and I'm comfortable with studying dense, more sophisticated texts as long as they are well written, comprehensive, and build the subject on the ground up. Online texts would be appreciated, as I'm spending the next few months travelling and wont be able to receive packages.

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Rigorous calculus? Then Calculus, by Tom Apostol: volume 1, volume 2, but don't buy it new, it's ridiculously expensive! I give the link only for the table of contents and the comments.

Rudin's Principles of Mathematical Analysis is also very nice.

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    $\begingroup$ I guess an important sub question that I forgot to ask is some advice on the distinctions between rigerous calculus and introductory analysis... I've got some sense of the distinction, and i have a feeling I'm more interested in analysis, but obviously I'm not experienced enough in either to decide. $\endgroup$
    – user3715913
    Sep 16, 2014 at 11:25
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    $\begingroup$ @user3715913 Honestly, I don't know for sure: in France, there is no "calculus", only analysis. And I have never seen a math course without rigor :-) I don't have Apostol at hand to check, but I believe it's more or less like the first two years of university, and it's more in the "analysis" side. Rudin is definitely an analysis book, with great and deep details. $\endgroup$
    – Jean-Claude Arbaut
    Sep 16, 2014 at 11:30
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    $\begingroup$ Non-rigorous calculus ... this is the course for science and engineering students. In the US, at small colleges (and high schools, as the OP) it would be the only calculus course. Mathematics students would continue with a rigorous course later, while most science and engineering students will not need more than this. $\endgroup$
    – GEdgar
    Sep 16, 2014 at 12:04

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