My goal in writing Understanding Analysis was to create a lively, one-semester introduction to real analysis that exposes students to the rich rewards inherent in taking a rigorous approach to the study of functions of a real variable. The first several times I taught such a course, my students became proficient at writing mathematical proofs, but I realized that the content of the course I had taught was essentially a long verification of the theorems of introductory calculus. In the end, it was hard to justify all the hard work they had invested. “Well,” I said, “it turns out that if you continue on, you discover that not every closed set is a union of closed intervals, most continuous functions are nowhere differentiable, and not every Taylor series converges back to the function that generated it. Derivatives, it turns out, satisfy the conclusion of the Intermediate Value Theorem but not the hypothesis, the Riemann integral can only handle a ‘small’ number of discontinuities, and, tragically, the Riemann integral cannot even integrate every derivative.”
Understanding Analysis outlines an introductory course in real analysis where, instead of saying “It turns out that…,” we actually address the issues directly. I honestly do not believe this makes the course any harder. The same list of core topics are treated here in roughly the usual order that they appear in most introductory modern treatments. The difference is where the emphasis is placed. We all know that the precision required in a rigorous convergence proof is difficult for students, but by shifting the focus of the exposition onto questions where the tools of analysis are really needed (e.g., Cantor sets, rearrangements of infinite sums, term-by-term differentiation of a series of functions), the hard work of a rigorous study is justified by the fact that these questions are inaccessible without it.
For over two decades, Understanding Analysis has been adopted at a wide range of colleges and universities, and also used effectively by individuals working on their own. A solutions manual is available and can be obtained from Springer by instructors who have adopted the text. If you are interested in having a copy for self-study, please contact me (abbott@middlebury.edu) directly.
