Limits of Functions
Definition of a Function
f
:
D
→
Y
Let
D
and
Y
be nonempty sets. A rule
f
that assigns to each member of
D
a
unique
member of
Y
is a function from
D
to
Y
.
We write the relationship between a member
x
of
D
and the member
y
of
Y
that
f
assigns to
x
as
y
=
f
(
x
)
.
The set
D
is the domain
of
f
, denoted by
D
f
. The members of
Y
are the
possible values
of
f
. The range
of
f
is
defined by
range of
f
:=
{
f
(
x
):
x
∈
D
} ⊂
Y .
Setup for rest of Handout
•
f
:
D
f
→
Y
is a function with
D
f
⊂
R
and
Y
⊂
R
.
•
x
0
∈
R
and
a
can be
x
0
,
x
+
0
,
x

0
, or
±∞
.
•
L
∈
b
R
:=
R
∪ {±∞}
.
Overview
Loosely speaking, we say that the limit of
y
=
f
(
x
) as
x
appoaches
a
is
L
, and we write
lim
x
→
a
f
(
x
) =
L ,
provided that the value of
f
(
x
) can be made as close to
L
as we wish by taking
x
sufficiently close to,
but not equal
to
,
a
.
We now need to make this concept precise! This is where the topology we learned comes in!! What does it mean to
be
close
? Well, that it is in a
basic neighborhood
. Let’s look at some of these (deleted)
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 Fall '10
 Girardi
 Limits, Sets, Limit, Continuous function, Equals sign, Neighbourhood

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