So, for instance, maybe Baby Rudin is the paradigm of "how to do it right" which includes proving stuff like Baire's Theorem. By "rigorous", I don't necessarily mean "Baby Rudin". On the other hand, Thomas and Finney does, in fact, have the formal definition of a limit, for instance, on p70 of my edition. But, that's after a couple sections covering limits and even theorems about limits, done as sort of a digression and a computational one at that, if you look at the exercises. Clearly, students aren't really expected to be able to do epsilon-delta proofs, and the material of the book doesn't appear to entail that level of sophistication. So, by "rigorous", I also don't mean that sort of thing, either -- where it is technically "in there" somewhere but not really taught. Apostol is not quite Real Analysis, yet he definitely has an impressive table of contents for a calculus book. For instance, if you look at the way he approaches integration, he starts out with simple finite sums of step functions and you get exercises on that. Then, he extends that using the Archimedean Property. If you go back to the section where he covers the Archimedean Property, he's got all kinds of nice exercises in there asking the student to prove stuff like that the rationals are dense in the reals. On the other hand, on that section for integration, those exercises are pretty much all computational. I don't think there is a single problem in that exercise set that asks the student to mimic his proof that a monotonic function on a closed interval is integrable or that otherwise requires the use of the Archimdean Property to figure out or do. Of course I don't want to judge a book on two sections -- I'm just looking for other people's opinion on the book. How rigorous is it? Is this really just Thomas and Finney on steroids? Is Courant, for instance (which I have just ordered a copy of but have not really been able to really look at), qualitatively different from this? Is this just calculus for you -- if I really want to teach rigor in this venue then I will just have to wait until real analysis to do it?