Unformatted text preview: r x p we have
log x .
Now choose a positive integer n such that
p log p n.
Since x log x n for x sufficiently large x, we can use the Bolzano intermediate value theorem to choose
a number a p such that a log a n. Now since
a log a a log a log a
a log a
we can use the Bolzano intermediate value theorem again to choose a number b
a, a such that
b log b n
We now observe that 231 |f a sin n
2 f b | sin n 1 and so the proof is complete.
5. a. A function f from a metric space X to a metric space Y is said to be Lipschitzian on a set S if there exists
a number k such that the inequality
d f t ,f x
kd t, x
holds for all points t and x in S. Prove that every lipschitzian function is uniformly continuous.
Suppose that f is a function from a metric space X to a metric space Y, that k is a positive
number and that the inequality
d f t ,f x
kd t, x
holds for all points t and x in the space X. Suppose that 0. We define /k and observe
that, whenever t and x belong to X and d t, x we have
kd t, x k d f t ,f x
b. Given that f x x for all x k . 0, 1 prove that f is uniformly continuous but not lipschitzian on 0, 1 . Solution: The fact that f is uniformly continuous on the closed bounded set 0, 1 follows at once
from the fact that f is continuous there. Now, to prove that f fails to be Lipschitzian, suppose that k is
any positive number. Given x
0, 1 we see that
|f x f 0 |
|x 0 |
and this exceeds k whenever x 1/k 2 . 6. a. Prove that if f is a uniformly continuous function from a totally bounded metric space X onto a metric
space Y then Y is also totally bounded.
We suppose that f is a uniformly continuous function from a totally bounded metric space X
onto a metric space Y. To show that Y is totally bounded, suppose that 0. Choose 0
such that whenever t and x belong to X and d x, t we have d f x , f t . Using the fact
that X is totally bounded, choose finitely many points x 1 , x 2 , , x n in the space X such that
n X B xj, .
j1 We shall now show that
n Y B f xj ,. j1 Suppose that y Y. Using the fact that the function f is onto the space Y, we choose a member
x of X such that y f x . Choose j such that x B x j , . We see that y B f x j , .
b. Give an example of a uniformly continuous function from a bounded metric space onto an unbounded
If X is the discrete space in which
d x, y 0 if x y
1 if x y then every function from X to another metric space is continuous.
c. Prove that if a function f is uniformly continuous on a bounded subset S of R k into a metric space Y then
the range of f is a bounded subset of Y.
We know that whenever S is a bounded subset of R k , the subspace S is totally bounded.
d. Give a quick proof that if f x 1/x for all x
interval 0, 1 . 232 0, 1 then f fails to be uniformly continuous on the Had this function been uniformly continuous then its range would have had to be bounded. But
its range i...
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