I have found that certain questions concerning the theoretical underpinnings of beginning (single-variable) calculus arise frequently among those who teach the first two or three calculus courses. The purpose of this book is to answer those questions, and others as well. Among those who raise the questions I have tried to answer will be the better students in elementary calculus classes. In my experience, many of their instructors are unsure, themselves, of the answers to those questions; I hope that both students and teachers will find the answers they seek here.
I have tried to make it an over-riding principle that all of the proofs I give here should be accessible, insofar as that is possible, to the outstanding students who are likely to be unsatisfied with their calculus text's dismissal of certain topics or arguments as "beyond the scope of this book". You will not now find that frustrating phrase in the body of this book, and I fully intend that you will not find it in the portions of the book that are yet unwritten. However, a reader may find it necessary to track the answer to a particular question back through answers to questions that precede the original in the logical structure of calculus. I have not tried to make the book self-contained. Elementary calculus texts do a reasonable job at the tasks they undertake, and this book is meant for use in conjunction with such a text. I have tried to answer the questions that the textbooks do not answer.
What I have posted here is a preliminary version of the preliminary version. As the work stands at this writing, it is far from complete. There are currently but seven chapters. I have covered perhaps a half of the things I intend to cover. Nevertheless, I believe that what is here can be of use to practicing teachers. Moreover, I am hoping for constructive comment from all who take the time to read it. Among other things, I am interested in knowing what burning questions you have that I didn't address, or that I did address--but poorly. Please address your comments to firstname.lastname@example.org. If you prefer quainter methods of communication, you may write to