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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 ΠX 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:
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