That's easy, Dr.Stillwell -or may I call you John?-
[original suggestion/answer by Andrew L]
Constantin Carathéodory's Vorlesungen über Reelle Funktionen. Why this book has never been translated into English is simply a mystery to me.
And while he's at it,let's it, let's get whoever's on that case to get Courant and Hurewitz's treatise on complex functions into English as well, so I can see finally if it's as good as Serge Lang always said it was........was...
2 last requests while I'm at it:Faddev's it: Faddeev's 1984 "Lectures In Algebra" and the second edition of Kostrikin's 3 volume "Introduction To Algebra"Algebra. I'm such a sucker for Russian texts, they're so beautiful and concrete with connections to physics. We Westerners can learn so much from their approach.