Preface
This book evolved from a series of lectures at the University of Sussex and is
designed to provide an integrated course in real and complex analysis for under-
graduates who have taken first steps in real analysis; the intention is to exhibit
something of the interplay between these and other areas of mathematical study.
The prerequisites are modest: it would be completely sufficient to have followed
preliminary courses in real analysis (involving e, d ideas) and algebraic structures.
There are many exercises, ranging from the elementary to the quite demanding.
To establish notation and terminology, some prerequisites are reviewed in the
appendices.
A persistent theme in the text is the search for a primitive. In the case of real
analysis, the Riemann integral offers one route in this quest and, with an eye to
complex analysis, the improper Riemann integral is an extension consonant with
the demands of contour integrals.
Chapter 1 deals with the Riemann theory of integration on the real line using
the simple and elegant approach due to Darboux that quickly leads to the basic
properties of the integral together with means of evaluation and estimation. It also
enables direct, elementary proofs to be given of the results that if f is
Riemann-integrable, then (i) the set of its points of continuity is dense in the
domain of f, and (ii) g f is Riemann-integrable if g is continuous. A character-
ization of the class of Riemann-integrable functions, from which these last two
assertions follow, is postponed to the next chapter as it is technically more chal-
lenging. The Riemann integral is confined to bounded functions defined on closed
bounded intervals and requires extension to cope with the demands of later
chapters. To allow for some relaxation of these constraints, the improper Riemann
integral is introduced. We indicate the limitations of the Riemann integral which
led to the development of Lebesgue’s integral (which itself would require slight
extension for use in the later chapters), of which the former is a special case.
Metric spaces form the theme of Chap. 2; the earlier one provides a wealth of
examples of such objects. Detailed coverage is given of the core properties of
completeness, compactness, connectedness and simple connectedness: this last
property is highlighted. While it has become more common in recent times to
present such matters in the context of normed linear spaces, we believe it is
important for the student to realize that linear structure is irrelevant to many of the
results. Regarding completeness, Cantor’s characterization is established as are
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