Efficiently organized; ideas introduced early are used later.
Many illustrative examples.
Generous exercise sets, including many routine exercises designed to de-
velop student confidence.
Every chapter begins with a rationale and suggestions for coverage.
Every chapter contains clearly identified project-type exercises that ad-
vance student knowledge beyond the level of this book.
Sequences are seen as a unifying theme, recurring as a useful tool through-
out the course.
Topological concepts and language are used extensively, because they help
unify the subject.
The Cantor set and Cantor's function are covered completely.
Exponential and logarithm functions are defined rigorously in three sepa-
rate contexts: in Section 5.6 in the context of continuous monotone func-
tions; in Section 7.7 using the integral; and in Section 8.8 using infinite
series. Similarly, trigonometric functions are defined and developed rigor-
ously using the integral in Section 7.7 and using series in Section 8.8.
Many surprising, even "pathological,'' examples appear throughout the
text and exercises. Certain functions appear in chapter after chapter,
forming a unifying chain of examples: Dirichlet-type functions, Thomae's
function, the absolute value and related functions, and relatives of sin(l/x).
Real analysis has historically derived much motivation from examples
such as these.
Many "applications" are shown to follow unexpectedly from the big ideas
of the course. For example, the irrationality of e is derived from Taylor's
theorem in Chapter 6; the irrationality of 7r and ex for rational x is derived
from the Fundamental Theorem of Calculus in Chapter 7.
The core of Chapters 1- 7 (material not labeled with a"") can be learned
by the typical student in a one-semester course. The "" material and
later chapters can be omitted in a one-term or one-semester course. Learn-
ing the "*" material will challenge the more talented student, and covering
the entire book will require a second semester.
Certain advanced topics are suitable for individual or group projects.
They are clearly identified and accompanied by appropriate guidelines.
A review of useful background material on logic, strategies of proof, sets,
and functions appears in Appendices A and B.
