Originally Posted by Bostonian
OK. After the Brown book I suggest a relatively short book

http://www.amazon.com/Analysis-Elementary-Functions-Robert-Sorgenfrey/dp/0395340578
Analysis of Elementary Functions
Robert H. Sorgenfrey and Edwin F. Beckenbach

that provides a bridge from trigonometry to calculus, and/or "Modern Introductory Analysis" by Dolciani et al (there are several editions).

This reminds me that I meant to say: I don't know where "proper" analysis, the sort I learned in the first year at university, gets taught in the US high school/university sequence, but I think your DD might really appreciate learning it alongside calculus, or even before, HowlerKarma. School calculus (and this certainly applies to the AP calculus syllabus, which I have looked at) tends to nod at continuity and the definition of derivatives as limits, and then expect you to get on with it by rote. Tom K"orner's notes here
https://www.dpmms.cam.ac.uk/~twk/C5.pdf
are a good guide to the stuff I'm talking about, and his "Further Reading" section at the end recommends some books.

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