6 Introductionone y. ~onversely, given such a subset H of X x Y, the single-vafued
function f : X ---? Y is then well defined (by assigning to each x the y that
is paired with that x ). On account of this, some authors define the function
to be the set H itself. Although there is no objection to this procedure from
a logical point of view, to consider a function as a set certainly runs at odds
with our intuitive idea of this concept and with its usage in applications.
What is more important, it leads in the theory of functions to an overly
elaborated symbolism and to some technical difficulties.
It should be noted that a set mapping f : G -+ P(Y), G C P(X), is
really a single-valued function, the elements in the domain and codomain
being subsets of X and Y, respectively. Thus the concept of a single-
valued function should be regarded as the fundamental one in the theory
of functions.
0.3 Notations
In addition to the set and mapping notations introduced above, the
following will be used consistently throughout the book:
The so-called decimal notation, due to G. Peano, is used to enumerate
consecutively sections, definitions, theorems, corollaries, equations, and
figures within each chapter. Thus Section 2.3 indicates the third section
of the second chapter, and similarly for theorems, corollaries, and figures.
Fo'l' equations (or formulas) we write, for example, 5.2-8 to indicate the
eighth equation of the second section of Chapter 5.
iff is often used to abbreviate the phrase "if and only if."
JR denotes the system of real numbers and JR+, the set of positive real
numbers including zero (i.e., the set of all nonnegative real numbers).
C indicates the system of complex numbers, and (',* = C U ( oo ), the·
extended complex number system.
An asterisk preceding a problem number indicates that the property to be
proved in that problem will be used later in the book and thus must be
considered as part of the text.
Special notations are introduced as needed.