Preliminaries

The typical reader of this text will have had a rigorous “δ- $$\varepsilon$$ ” first year calculus course, using a text such as for example [22]. Such a course will have included definitions and basic properties of the standard elementary functions (polynomials, rational functions, exponentials and logarithms, trigonometric and cyclometric functions), the concept of continuity of a real function and the fact that continuity is preserved under standard constructions (sum, product, composition, etc.), and the basic rules of computing derivatives. We review here mainly the more theoretical aspects of these topics. The reason for reviewing them are two-fold. The first reason is that we would like this text to be as self-contained as possible. The second reason is that some of the basic results have, in fact, substantial depth in them, and the more advanced topics on which this book focuses make heavy use of them. Not reviewing such topics would at times even create a danger of circular arguments.