What is a short book which covers all of the key results in calculus in a rigorous manner?
I think Serge Lang's A First Course in Calculus does a pretty good job of that. However, I read maybe 200 pages of the book in a sitting back when I studied AP Calc, so I may have misjudged how rigorous the book is. I do remember that Lang presents I think motivation for a part of the proof of the chain rule in the chapter on differentiation, saving a harder case of the proof for the end of the chapter or something.
Also, I think the epsilon-delta stuff is in the appendix of the book. The main theorems about continuous functions: IVT, boundedness, attains max and min are not in the body of the book, but they may be in the appendix.
There's a book called "Introduction to Analysis" by Maxwell Rosenlicht which covers quite abit of real analysis and is quite small/short (it's also only $20). However, I would only recommend it if you're already comfortable with a lot of concepts in analysis because it's not amazingly expository since it's trying to be so condensed.
I doubt you'll find a concise book which is "rigorous" as well. If you want to look for a short, crash course in calculus the books will most likely be a review of the definitions and the techniques rather than the rigor presented in regular textbooks.
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