Classical analysis is rooted in differential and integral calculus. Calculus
began with Newton and Leibniz as a collection of ideas not precisely for-
mulated, but highly effective in scientific work. As the years progressed,
however, mathematicians recognized the limitations of intuitive concepts
and sought to place calculus on a firm theoretical foundation. Their efforts
led to a framework of definitions, theorems, and proofs that has become part
of the standard curriculum for students of advanced calculus.
This preliminary chapter offers a rapid overview of the basic theory of
calculus. Definitions and major theorems are stated for later reference, and
proofs are generally included. It is assumed that the reader is already ac-
quainted with most of this material but may feel a need for review. The
coverage is not complete, as some relevant topics are omitted. More exten-
sive treatments can be found in introductory texts such as Rudin , Ross
, or Mattuck .
1.1. Mathematical induction
Before turning to the calculus, we want to give a brief review of mathe-
matical induction. This is an important method for verifying relations that
depend on positive integers. The positive integers, also known as the natural
numbers, are simply the numbers n = 1, 2, 3,... . The symbol N will denote
the set of all positive integers.
Suppose now that Pn is a proposition expressed in terms of an integer
n, which is to be proved for all n ∈ N. The first step is to verify that P1
is true. This is usually quite trivial. Next comes the inductive step. Under