Understanding Analysis Undergraduate Texts in Mathematics by Stephen Abbott
ISBN 978144I928665, 9780387215068
My primary goal in writing Understanding Analysis was to create an elementary
one-semester book that exposes students to the rich rewards inherent in
taking a mathematically rigorous approach to the study of functions of a real
variable. The aim of a course in real analysis should be to challenge and improve
mathematical intuition rather than to verify it. There is a tendency,
however, to center an introductory course too closely around the familiar theorems
of the standard calculus sequence. Producing a rigorous argument that
polynomials are continuous is good evidence for a well-chosen definition of continuity,
but it is not the reason the subject was created and certainly not the
reason it should be required study. By shifting the focus to topics where an
untrained intuition is severely disadvantaged (e.g., rearrangements of infinite
series, nowhere-differentiable continuous functions, Fourier series), my intent
is to restore an intellectual liveliness to this course by offering the beginning
student access to some truly significant achievements of the subject.
