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functions: specification and definition
This section describes precisely what is meant by “function”. This involves complications because there are two different definitions of function in common use, even though this difference generally causes little trouble!
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We start with giving a specification of function, then get into the technicalities of the definitions.
Contents
Mathematical definitions of function
Specification of function
I will use two running examples throughout this discussion:
¨ F is the function defined on the set as follows:
.
This is the function called “Finite” in the
chapter on examples of functions.
¨
G is the real-valued function
defined by the formula .
Specification: functions
A function (
is the letter phi) is a mathematical object
which determines
and is completely determined by the following data:
¨
has a domain, which
is a set. The domain may be denoted by dom
.
¨
has a codomain,
which is also a set and may be denoted by cod
¨
For each element a of the domain of ,
has a value at a.
§
The value of at
a is completely determined by a and F .
§
The value of at
a must be an element of the codomain
of F.
§
The value of at
a is denoted by
(a).
The domain and codomain of a function are not always specified in a math text. More about that here.
Examples
¨ The definition of the finite function F
specifies that the domain is the set . Its codomain is not specified, but must
include the set {1,2,3}. The value of F at 3, for example, is 2, because the definition says that
F(2) = 3.
¨
The definition of G above does not specify the domain or the
codomain. The convention in the case of
functions on the real numbers is to
take the domain to be all
real numbers at which the formula is defined. In this case, that is every real
number, so the domain is . The codomain must include all real numbers greater than or equal to 4.
(Why?)
¨ The Split function is explicitly given the domain the closed interval [0,1] . Its codomain is not given. It must include [0,1].
¨
The Sine Blur function is
explicitly given the domain of positive real numbers. Its codomain must include the closed interval
[1,
1] since the sine function takes every value in that interval.
Terminology incomplete
Map, mapping
Some texts use this word interchangeably with the word "function”. Others distinguish between the two, for
example requiring that a mapping be a continuous function.
Functional
Multivalued function
Partial function
Notation
Ways of giving a function
By formula, algorithm, etc
Arrow Notation
¨
In the expression " (a)":
§ a is called the argument or independent variable or input to F.
§
(a) is the value or dependent
variable or output. (See dependency relation.)
¨
The operation of finding (a)
given
and a
is called evaluation or application.
Examples
For we could write
(choosing the codomain to be all of
). For
the finite
function we could write
.
Usage
The expression has context-sensitive pronunciation.
¨
Standing
alone, the expression may be read aloud this way: “ is a function from A to B”
or “
goes from A
to B”.
¨
The expression may occur embedded
in a sentence, as in “Let be a differentiable function.” This may be read “Let
from A to B be a
differentiable function.”
Warnings
¨
The statement by itself does not determine the function
. It
says only that its name is
,
its domain is the set A, and its codomain
is the set B. For example, for
,
and a gazillion other functions we may have
.
¨
You should distinguish between , which is the name of the function and
(a),
which is the value of
at an input value a. Nevertheless, such a
function is very commonly referred to as
(x). For functions given by formulas, this notation
has the value of telling you what letter will be used for the input variable.
Mathematical definitions of function
This section gives the definition(s) of function usually given in textbooks.
In the nineteenth century, mathematicians realized that it
was necessary for some purposes (particularly harmonic analysis) to give a mathematical definition
of the concept of function. A stricter
version of this definition turned out to be necessary in algebraic topology and
other fields, so now there are two
nonequivalent definitions in common use.
The difference
between these definitions causes much less trouble than you would think!
To state these
two definitions we need a preliminary idea.
The functional property
Definition
A set R of ordered pairs has the functional property if two pairs in R with the same first coordinate have to have the same second coordinate (which means they are the same pair).
Examples
¨ The