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Unformatted text preview: 1 , x 2 we have
.
d g f x1 , g f x2
11. Suppose that f is a continuous function on a subset S of a compact metric space X. Prove that the following
two conditions are equivalent:
a. The function f is uniformly continuous on S.
b. It is possible to extend f to a continuous function on the set S.
The fact that condition a implies condition b follows at once from Exercise 7. Since S is compact, a
continuous function on S must be uniformly continuous. Therefore, if f has a continuous extension
to S, then f must be uniformly continuous on S.
12. Is it true that if f is a uniformly continuous function from a bounded metric space X onto a metric space Y
then Y is bounded?
Not a chance! If X is the discrete space in which
0 if x y d x, y 1 if x y then every function from X to another metric space must be uniformly continuous.
13. Is it true that if f is a uniformly continuous function from a complete metric space X onto a metric space Y
then Y is complete?
No, this statement is false. For example, if we define
1
fx
1 x2
for x
0, Ý then f is a uniformly continuous function from 0, Ý onto 0, 1 .
14. Suppose that f is a oneone uniformly continuous function from a metric space X onto a metric space Y and
that the inverse function f 1 of f is continuous on Y. Prove that if Y is complete then so is X.
Suppose that x n is a Cauchy sequence in the space X. Since f is uniformly continuous from X to
Y, it is easy to see that the sequence f x n is a Cauchy sequence in the space Y. Since Y is
complete, the sequence f x n must converge. We define
y n Ý f xn .
lim
Since the function f 1 is continuous at the point y we have f 1 f x n
f 1 y as n Ý. In other
1 y as n
words, x n f
Ý and we have shown that the sequence x n is convergent.
15. True or false: If f is a uniformly continuous oneone function from a complete metric space X onto a
complete metric space Y then the inverse function f 1 of f must be continuous.
No, this statement is false. We take X 0, 1 Þ 2, Ý and we define
fx x
1 235 if 0
1
x x if x 2 1 1.4
1.2
1
0.8
0.6
0.4
0.2
0 1 2 3 4
x 5 6 7 8 The above figure also shows the line y 1 in red. We see that f is a uniformly continuous oneone
function from the complete space 0, 1 Þ 2, Ý onto the complete space 0, 3 .
2
16. We shall say that a subset S of a metric space X is compressed if for every number
different points x and t in S such that d t, x . 0 there exist two a. Prove that every compressed subset of a metric space must be infinite.
This statment is obvious because any finite set positive numbers has a least member that is
positive.
b. Prove that if a subset S of a metric space X has a limit point then S must be compressed.
Suppose that S is a subset of a metric space X and that S has a limit point x. Suppose that
0. Choose a point u B x, /2
x . Now, using the fact that the number d x, u is positive,
choose a point v B x, d x, u
x . Since both u and v belong to the ball B x, /2 we have
d u, v .
c. Give an example of...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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