Introduction 5
have h( x) = Yo for all x E X, the function h : X -+ Y is called a constant
function in X.
If to each singleton { x} E G 1 corresponds under l a set B E 'P(Y), B not
a singleton, then the mapping l : G 1 -+ 'P(Y) is called a multiple-valued
function (also, a many-valued function or a multifunction). As before, theset Bis called the image of {x} under l, and we write B = l(x). When Y
is a set of numbers, in case of a multiple-valued function it is not possible
to combine the "values" l(x) by means of the ordinary rules of algebra,
but in certain cases we may apply to them the rules of the algebra of sets
(discussed in Chapter 2), or other special rules (see, e.g., Section 1.13).
However, for the multiple-valued functions arising in complex analysis it
happens that as {x} varies in G1 (or x in X), each Yk EB varies in some
Bk CB, with the Bk overlapping only at some exceptional points, so it is
possible to separate l into a countable number of single-valued functions
fk (called branches of l), and then operate with the values lk(x),k fixed,
as in ordinary algebra.
A single-valued function l : X -+ Y induces the following set mappings.
First, if A C X, we have
l(A) = {l(x): x EA} (0.2-1)
Second, if B CY, the inverse image of B, denoted l-^1 (B), is the subset
of X defined by
l-^1 (B) = {x: x EX, l(x) EB} (0.2-2)
The inverse mapping 1-^1 need not be a single-valued function;-i.e., for
y E Y, l-^1 (y) need not be a singleton (in fact, in most cases it is not). For
example, if x E ~and f(x) = x^2 , we have l-^1 (x^2 ) = {x,-x}.
Any mapping l : X -+ Y (not necessarily a function) induces also an
inverse mapping 1-^1 : Y -+ X, which is defined by the same expres-
sion (0.2-2). It may happen that l-^1 (B) = 0 for some B C Y, if there
is no x E X such that l(x) EB.
The mapping defined by a single-valued function l is said to be one-to-one (briefly, 1-1), or injective, if f(x) = l(x^1 ) implies that x = x^1 • If the
mapping is both surjective and injective (i.e., onto and one-to-one), it is
called bijective. In the latter case the inverse mapping is also a single-valued
function.
A single-valued function l : X -+ Y determines a subset H of X x Y
defined by