Preface to analysis of functions of a single variable: a detailed

For Christy My Light

I have written this book primarily for serious and talented mathematics scholars, seniors or first-year graduate students, whoby the time they finish their schooling should have had the opportunity to study in some detailthe great discoveries of our subject. What did we know and how and when did we know it?I hope this book is useful toward that goal, especially when it comes to the great achievements of that part of mathematics known as analysis.I have tried to write a complete and thorough account of the elementary theories of functions of a single real variableand functions of a single complex variable. Separating these two subjects does not at all jive with their development historically,and to me it seems unnecessary and potentially confusing to do so. On the other hand, functions of several variables seems to meto be a very different kettle of fish, so I have decided to limit this book by concentrating on one variable at a time.

Everyone is taught (told) in school that the area of a circle is given by the formula $A=\pi r^2.$ We are also told that the product of two negatives is a positive, that you cant trisect an angle, and that the square root of 2 is irrational.Students of natural sciences learn that $e^{i\pi }=-1$ and that ${sin}^2+{cos}^2=1.$ More sophisticated students are taught the Fundamental Theorem of calculus and the Fundamental Theoremof Algebra. Some are also told that it is impossible to solve a general fifth degree polynomial equationby radicals. On the other hand, very few people indeed have the opportunityto find out precisely why these things are really true, and at the same time to realize just how intellectually deep and profound these “facts” are.Indeed, we mathematicians believe that these facts are among the most marvelous accomplishments of the human mind.Engineers and scientists can and do commit such mathematical facts to memory, and quite often combine them to useful purposes.However, it is left to us mathematicians to share the basic knowledge of why and how,and happily to us this is more a privilege than a chore. A large part of what makes the verification of such simple sounding andelementary truths so difficult is that we of necessity must spend quite a lot of energy determining what the relevant words themselves really mean.That is, to be quite careful about studying mathematics, we need to ask very basic questions: What is a circle? What are numbers?What is the definition of the area of a set in the Euclidean plane? What is the precise definition of numbers like $\pi ,$ $i,$ and $e?$ We surely cannot prove that $e^{i\pi }=-1$ without a clear definition of these particular numbers. The mathematical analysis story is a long one, beginning with the early civilizations, and in some sense only coming to a satisfactory completion in thelate nineteenth century. It is a story of ideas, well worth learning.