This edition differs from the original 2001 version only in that we corrected a number of misprints and other errors. We are grateful to the many users of that version for notifying us of errors they found. We would like to make special mention of Richard Delaware (University of Missouri-Kansas City), and Steve Agronsky (California State Polytechnic University, San Luis Obispo), both of whom went through the entire first edition, made many helpful suggestions, and found numerous errors.

University mathematics departments have for many years offered courses with titles such as Advanced Calculus or Introductory Real Analysis. These courses are taken by a variety of students, serve a number of purposes, and are written at various levels of sophistication. The students range from ones who have just completed a course in elementary calculus to beginning graduate students in mathematics. The purposes are multifold:

1. To present familiar concepts from calculus at a more rigorous level.
2. To introduce concepts that are not studied in elementary calculus but that are needed in more advanced undergraduate courses. This would include such topics as point set theory, uniform continuity of functions, and uniform convergence of sequences of functions.
3. To provide students with a level of mathematical sophistication that will prepare them for graduate work in mathematical analysis, or for graduate work in several applied fields such as engineering or economics.
4. To develop many of the topics that the authors feel all students of mathematics should know.