> Do you or anyone else know a book that mentions/explains the > "real and all encompassing" definition of the derivative (well, > a book that explains REAL CALCULUS on the whole would be better, > by real I don't mean the number set). I don't care if this book > is really meant for 2nd or higher year students, because as far > as I'm concerned, learning without truly understanding is a waste > (my Calculus 1 class!!).
Michael Spivak, CALCULUS, 3'rd edition, Publish or Perish, 1994, xiv + 670 pages.
Tom Apostol, CALCULUS VOLUME 1: ONE-VARIABLE CALCULUS WITH AN INTRODUCTION TO LINEAR ALGEBRA, 2'nd edition, Blaisdell Publishing Company, 1967, xx + 666 pages.
Richard Courant and Fritz John, INTRODUCTION TO CALCULUS AND ANALYSIS, Volume 1, 1965, xxiv + 661 pages. [This has been recently reprinted, I believe.]
Spivak and Apostol are widely used in honors classes and at places like CalTech, MIT, Univ. of Toronto, etc. Courant/John is a classic that is in some ways pitched at an even higher level (the problems are harder, for one thing), but it gives an outstanding coverage of *everything*, from theoretical to applied, that someone planning to continue studying mathematics ideally should be exposed to in an elementary calculus course.
Incidentally, Apostol and Courant/John each have a 2'nd volume covering multivariable calculus, but I think for what you're looking for the 1'st volumes will suffice.
Although the three books above are probably the best fit for you, you might also want to look at the following.
Victor Bryant, YET ANOTHER INTRODUCTION TO ANALYSIS, Cambridge University Press, 1990, viii + 290 page.
All exercises have solutions. Full of pictures, diagrams, and other pedagogical aids. Excellent as a direct sequel to the standard calculus sequence and the book is carefully written with this in mind.
George R. Exner, INSIDE CALCULUS, Undergraduate Texts in Mathematics, Springer-Verlag, 2000, xviii + 211 pages.
Ernst Hairer and Gerhard Wanner, ANALYSIS BY ITS HISTORY, Springer-Verlag, 1996, x + 374 pages.