This book is first of all designed as a text for the course usually called "theory of functions of a real variable". This course is at present
customarily offered as a first or second year graduate course in United
States universities, although there are signs that this sort of analysis
will soon penetrate upper division undergraduate curricula. We have
included every topic that we think essential for the training of analysts,
and we have also gone down a number of interesting bypaths. We hope
too that the book will be useful as a reference for mature mathematicians
and other scientific workers. Hence we have presented very general and
complete versions of a number of important theorems and constructions.
Since these sophisticated versions may be difficult for the beginner, we
have given elementary avatars of all important theorems, with
appropriate suggestions for skipping. We have given complete definitions,
explanations, and proofs throughout, so that the book should be usable
for individual study as well as for a course text.