This book is a basic text in advanced calculus, providing a clear and well-motivated, yet precise and rigorous, treatment of the essential tools of mathematical analysis at a level immediately following that of a first course in calculus. It is designed to satisfy many needs; it fills gaps that almost always, and properly, occur in elementary calculus courses; it contains all of the material in the standard classical advanced calculus course; and it provides a solid foundation in the "deltas and epsilons" of a modern rigorous advanced calculus. It is well suited for courses of considerable diversity, ranging from "foundations of calculus" to "critical reasoning in mathematical analysis." There is even ample material for a course having a standard advanced course as prerequisite. Throughout the book attention is paid to the average or less-than-average student as weIl as to the superior student. This is done at every stage of progress by making maximally available whatever concepts and discussion are both relevant and understandable. To illustrate: limit and continuity theorems whose proofs are difficult are discussed and worked with before they are proved, implicit functions are treated before their existence is established, and standard power series techniques are developed before the topic of uniform convergence is studied. Whenever feasible, if both an elementary and a sophisticated proof of a theorem are possible, the elementary proof is given in the text, with the sophisticated proof possibly called for in an exercise, with hints. Generally speaking, the more subtle and advanced portions of the book are marked with stars (*), prerequisite for which is preceding starred material. This contributes to an unusual flexibility of the book as a text.
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