Graphics Reference
In-Depth Information
in
Y
and
U
= f
-1
(
O
q
) is open in
X
. Therefore, there is a d>0, so that
O
p
=
B
m
(
p
,d) «
X
Õ
U
. Clearly, f(
O
p
) Õ
O
q
. In other words, f is continuous at
p
. We are done.
4.2.6. Proposition.
The composition of continuous maps is continuous.
Proof.
The proof is easy using the second or third definition of continuity.
Definition.
If a function f :
X
Æ
Y
is not continuous at a point
p
but
()
L
=
lim f
q
qp
Æ
exists, then the point
p
is called a
removable discontinuity
of the function f (because
we could make f continuous at
p
simply by redefining f to equal
L
at
p
). Any other
point
p
where f is discontinuous is called an
essential discontinuity
.
For example, consider the functions f, g :
R
Æ
R
defined by
()
=
()
=
fx
1
,
for
x
π
0
,
f
0
2
,
and
()
=
()
()
=
g x
sin
1
x
,
for
x
π
0
,
g
0
2
.
The function f has a removable discontinuity at 0, whereas the function g has an essen-
tial discontinuity there.
If a function f :
X
Æ
Y
is continuous it is continuous at every point. If e>0, then
the first definition of continuity gives us a d>0 so that points within d of a point
p
will get mapped to a point within e of f(
p
). However, it is important to realize that the
d depends on
p
. It may
change
from point to point. A nice situation is one where one
can choose a d that will work for
all
points.
Definition.
A function f :
X
Æ
Y
is said to be
uniformly continuous
if for every e>0
there is a d>0 so that for all
p
,
q
Œ
X
with |
q
-
p
|<dwe have that |f(
q
) - f(
p
)|<e.
Uniform continuity is a very important concept in analysis.
4.2.7. Example.
It is not hard to show that the function
(
]
Æ
()
=
f
:
01
,
R
,
f x
1
x
,
is continuous but
not
uniformly continuous. For a fixed e, the d that works at a point
x gets smaller and smaller as x approaches 0. The core problem is that (0,1] is not
compact.
4.2.8. Theorem.
If f :
X
Æ
Y
is a continuous function and if
X
is compact, then f is
uniformly continuous.
Proof.
See [Buck78] or [Eise74]. See also Theorem 5.5.13.
The following property is weaker than uniform continuity but is sometimes
adequate: