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sequence u n in S converging to u and a sequence v n in S converging to v. Using the facts
that u n u and v n v and f u n
f u and f v n
f v as n Ý, choose N such that
whenever n N we have 233 d f un , f u d f vn , f v 3 3
d un, u
3
and d v n , v .
3 Since
d uN, vN d u N , u d u, v d v, v N
3
3
3 and therefore
d f uN , f vN
We conclude that
d f u ,f v d f uN , f vN 3 . d f vN , f v
.
3
3
3
Second Proof. (less messy but a little too slick, perhaps) Using the fact that f is uniformly
continuous on S, choose 0 such that whenever u and v belong to S and d u, v we have
d f u , f v . Now suppose that u and v are any points of S satisfying the inequality
d u, v . Choose a sequence u n in S converging to u and a sequence v n in S converging
to v. Since u n u and v n v and f u n
f u and f v n
f v as n Ý, we see that
d un, vn
d u, v and d f u n , f v n
d f u , f v . Using the fact that d u, v , choose N
such that the inequality d u n , v n holds whenever n N. Thus, for all n N we have
d f u n , f v n and therefore
.
d f u , f v n Ý d f un , f vn
lim
d f u , f uN 8.
a. Given that S is a set of real numbers, that a S S and that
f x x1a
for all x S, prove that f is continuous on S but not uniformly continuous. (Use this exercise.)
The result follows at once. b. Given that S is a set of real numbers and that S fails to be closed, prove that there exists a continuous
function on S that fails to be uniformly continuous on S.
The result follows from part a.
c. Is it true that if S is an unbounded set of real numbers then there exists a continuous function on S that
fails to be uniformly continuous on S?
No, it isn’t true. Every function defined on the set Z of integers is uniformly continuous there.
9. Given that f is a function defined on a metric space X, prove that the following conditions are equivalent:
This exercise was left in accidentally after being elevated to the status of a theorem. It appears as
Theorem 8.15.4.
a. The function f fails to be uniformly continuous on the space X.
b. There exists a number 0 and two sequences t n and x n in X such that d x n , t n
d f xn , f tn 0 as n Ý and for every n.
10. Is it true that the composition of a uniformly continuous function with a uniformly continuous function is
uniformly continuous?
Yes, it’s true. Suppose that f is a uniformly continuous function from a metric space X to a metric
space Y and that g is a uniformly continuous function from Y to a metric space Z. To show that the
composition g f is uniformly continuous, suppose that 0. Using the fact that g is uniformly 234 continuous on the space Y, choose 0 such that, whenever y 1 and y 2 belong to Y, and
d y 1 , y 2 , we have d g y 1 , g y 2 . Now, using the fact that f is uniformly continuous on the
space X, choose 0 such that, whenever x 1 and x 2 belong to X and d x 1 , x 2 , we have
d f x 1 , f x 2 . Then, whenever x 1 and x 2 belong to X and d x...
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 Fall '08
 STAFF
 Math, Calculus

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