##
Understanding Analysis, 2nd edition

#### Stephen Abbott, Middlebury College

####
Springer,
New York, 2015

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 to them the reasons for 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
in the integrand, 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 nearly two decades,*Understanding Analysis *has been adopted at a wide range of colleges and univesities, and also used effectively by individuals working on thier 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.

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