Preface Preface to the Second Edition
You are viewing the second edition of our open–source book which was originally titled, How We Got from There to Here: A Story of Real Analysis. As much as we liked our original title we discovered, much to our dismay, that it led many people to believe that it is about the history of mathematics rather than the real analysis textbook that it is. When we wrote it we assumed that our intention to use history to tell the story behind the nonintuitive, rigorous definitions and theorems of real analysis would be clear from the title. We were wrong. So we have changed the title for this second edition. We hope the new title clarifies our intent.
When we wrote the first edition we deliberately chose to forego any mention of integration theory in order to keep the size of the text manageable. We assumed that students were familiar with the methods by which integrals can be computed (especially Integration by Parts) from calculus class and we relied on this knowledge. But in keeping with our intention to follow the historical development of our topic we did not examine any underlying theory of integration. After all, integration was understood as antidifferentiation long before Cauchy, Riemann, Darboux, or Lebesgue were born.
After teaching from the first edition for many years, we came to the conclusion that we should really provide at least an introduction to the theory of integration for those students who might go on to a more advanced course.
To that end we have re–arranged some of the later topics as follows:
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We created a new chapter 10 (Limits, Derivatives, Integrals, and the Fundamental Theorem of Calculus) and placed in it the sections on the definitions of limit and derivative which had previously been in Chapter 8.
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We also added two new sections to Chapter 10. In Section 10.3 we prove the Fundamental Theorem Of Calculus and in Section 10.4 we define the Cauchy, Riemann, and Darboux integrals. If this seems backwards to you, consider that this is how mathematics is usually created. Foundational issues are typically the last piece of the puzzle to be put in place.
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Additionally we have expanded Chapter 12 (formerly Chapter 11) to include Section 12.1 and Section 12.3 where we define the Lebesgue integral in its historical context.In Section 12.1 we expand more on Cantor’s influence and the transferal of analysis to a modern set theoretical setting which provides the interested student with a path to studying point set topology in a future course. We believe this is appropriate for an introductory book.Of course, the rise in set theory leads to Lebesgue measure and the Lebesgue integral. We scratch the surface of those topics in Section 12.3 in order to provide the interested student with a starting point to begin studying those topics as well.
On a more nuts and bolts level, most of the problems in the last edition were embedded in the text but there were a few sections labeled “Additional Problems” at the end of some chapters as is tradition. But we firmly believe that the problems we give our students should be located in context within a textbook so in this edition we have absorbed those additional problems into the body of the text in the correct context. This is a result of our own evolution. After learning and teaching from conventional math books for most of our careers we have concluded that the problems we assign to our students are part of the story of our topic and should be treated as such. In particular, placing them at the end of a chapter or section obfuscates the point by forcing the student to spend time searching back through the text for the relevant discussion.
