Then is not defined, because

Let
be defined by the rule: for all

Thus , , ,

(3.40) |

(3.41) |

and

for all . Also

We call the

for example is the same as . The notation in formula (3.44) is always ambiguous. I will use it for sequences like

in which it is somewhat complicated to give an analytic description for .

If is a sequence, and
, then we often denote by .

For each let

and

denote the polygons inscribed in and containing described in section2.1. Then

and are sequences in .

is a real sequence. (Cf. (2.3) and (2.12).)

is a sequence of intervals.

The rules and are equal if and only if = for all . If and then it is customary to write to mean . This is an abuse of notation, but it is a standard practice.

and is defined by the rule

then since and have different codomains.

If
and
are defined by the
rules

then .

In certain applications it is important to know the precise codomain
of a function, but in many applications the precise codomain is not
important, and in such cases I will often omit all mention of the
codomain. For example, I might say ``For each positive number , let
.'' and proceed as though I had defined a function. Here
you could reasonably take the codomain to be
the set of real intervals, or the set of closed intervals, or the set of
all subsets of **R**.

is called the

We call the -image of . Clearly, for every subset of we have .

then

.

If the domain and codomain of are subsets of

The graph of is sketched below. The arrowheads on the graph are intended to indicate that the complete graph has not been drawn.

Let
Let be the function
from to **R** defined by the rule

The graph of is sketched above. The solid dot at indicates that is in the graph. The hollow dot at indicates that is not in the graph.

Let
be defined by the rule

Thus and The graph of is sketched above.

The term function (*functio*) was introduced into mathematics
by Leibniz [33, page 272 footnote]. During the
seventeenth century the ideas of function and curve were usually
thought of as being the same, and a curve was often thought of
as the path of a moving point. By the eighteenth century the
idea of function was associated with ``analytic expression''.
Leonard Euler (1707-1783) gave the following definition:

A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.

Hence every analytic expression, in which all component quantities except the variable are constants, will be a function of that ; Thus ; ; ; ; etc. are functions of [18, page 3].

The use of the notation ``'' to represent the value of
at was introduced by Euler in 1734 [29, page 340].

**a)**- .
**b)**- .
**c)**- .
**d)**- .