Preface xviiChapter 4 focuses on limits of functions. The c-6 techniques are discussed
early and used extensively. The sequential criterion is an integral part of the
presentation. One-sided limits and the use of infinity in limits a re given full
treatment, but the instructor may want to skim lightly over these sections,
which the average student can handle as reading assignments.
Chapter 5 on continuous functions is a meaty chapter. The c-6 techniques
are discussed in full detail, and the methodology of sequ ences is shown to b e
a powerful supporting tool. One-sided continuity, discontinuities, and implica-
t ions for monotone functions are discussed. Special emphasis is given to two
powerful results about continuous functions: the continuous image of a com-
pact set is compact, and the continuous image of an interval i s an interval.
Uniform continuity is treated in Section 5.4, but in a one-semester course this
topic can be postponed until it is needed in Chapter 7 for proving t hat contin-
uous functions are Riemann integrable.
A novel feature of this book is the (optional) "early" treatment of ex-
ponential and logarithm functions, in line with the trend to introduce these
functions early in the calculus sequence. Thus, Section 5.6 provides a rigorous
development of these functions and is somewhat novel in this regard. In order
to provide the necessary b ackground for these functions I felt it necessary to
p recede this material with a section on monotone functions, continuity, and
inverses. This, in turn, provided the p erfect opportunity to discuss Cantor's
function. Chapter 5 ends with t he Baire category theorem, and a proof that
the set of discont inuities of a real function must be an Fa set.
Chapter 6 is a comprehensive treatment of differentia bility and the deriva-
tive. All the usu al rules for differentiation are proved rigorously. This includes
rules for general power, exponential, and logarithm functions, which were intro-
duced in Chapter 5. The relationship b etween the derivative, monotonicity, and
extreme values is explored carefully, as is the intermediate value property of
derivatives. Rolle's theorem, the mean value theorem, and many of their appli-
cations, are given full attention. Taylor's theorem is seen as a mean-value-typ e
theorem, and L'Hopital's rule is derived from Cauchy 's mean value theorem.
In Chapter 7 the Riemann integral is defined using the Darboux sum ap-
proach. It is my view that, for students at this level, this is still the most ap-
propriate integral and its most natural definition. Riemann's criterion is proved
and used to establish the integrability of monotone and continuous functions.
The integral is shown to be a limit of Riemann sums, and the first and second
funda mental theorems of calculus are proved. A novel feature is the proof that
regul.ar partitions are sufficient for Riemann integrability. In an optional section
the exponential, logarithmic, and trigonometric functions are defined and de-
veloped using the integral. In the final (optional) section, Leb esgue's criterion
for Riemann integrability is proved.
Chapter 8 is a standard presentation of the theory of infinite series of
real numbers, including Cauchy product series, the theory of power series, real
analytic functions, and a little on double series. All t he customa ry convergence