Graphics Reference
In-Depth Information
()
-< .
f
qL
The function is said to be
continuous
at
p
if
()
=
()
lim
f
qp
f
.
qp
Æ
This leads to the classical definition of a continuous function.
First definition of a continuous function.
The function f :
X
Æ
Y
is said to be
continuous
if it is continuous at every point of
X
.
There are conceptually better and cleaner definitions. The motivation for our new
definitions lies in the fact that one can rewrite the definition of continuity of a func-
tion f at a point
p
as follows:
(
)
Õ
m
(
)
«
n
(
()
)
«
For every
e
>
0
there is a
d
>
0
so that f
Bp
,
d
X B p
f
,
e
Y
.
(4.1)
Second definition of a continuous function.
A function f :
X
Æ
Y
is said to be
continuous
if f
-1
(
V
) is open in
X
for every open set
V
in
Y
.
Third definition of a continuous function.
A function f :
X
Æ
Y
is said to be
continuous
if f
-1
(
V
) is closed in
X
for every closed set
V
in
Y
.
The second and third definitions are the “right” definitions, which extend to
abstract topological spaces as will be seen in Chapter 5. The second is actually the
most common. Since the next theorem shows that the three definitions are in fact
equivalent, we shall not distinguish between them.
4.2.5. Theorem.
The three definitions of continuity are equivalent.
Proof.
We show that the first and second definitions are equivalent and leave the
rest as an exercise for the reader. Assume that f is continuous with respect to the
first definition. Let
V
be an open set in
Y
. We need to show that
U
= f
-1
(
V
) is open in
X
. See Figure 4.5. Let
p
be any point in
U
and let
q
= f(
p
). Choose e>0 so that
O
q
=
B
n
(
q
,e) «
Y
Õ
V
. By statement (4.1), there is a d>0 so that f(
O
p
) Õ
O
q
, where
O
p
=
B
m
(
p
,d) «
X
. Therefore,
p
belongs to an open subset
O
p
of
X
that is contained
in
U
. Since
p
was an arbitrary point of
U
, we have shown that
U
is an open set in
X
.
Conversely, assume that f is continuous with respect to the second definition of
continuity. Let
p
Œ
X
and set
q
= f(
p
). Let e>0. Then
O
q
=
B
n
(
q
,e) «
Y
is an open set
X
Y
f
U = f
-1
(V)
V
q
p
Figure 4.5.
Understanding continuous
functions.
