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f H of f fails to be closed in R.
We can define f x 1 for x 1. This function f is continuous on the closed set 1, Ý and its range
x
is 0, 1 which is not closed.
2. Give an example of a function f that is continuous from a closed set H of real numbers into R such that range
f H of f fails to be bounded.
We can define f x x for every x R.
3. Give an example of a function f that is continuous from a bounded set H of real numbers into R such that
range f H of f fails to be closed in R.
We can define f x x for 0 x 1.
4. Give an example of a function f that is continuous from a bounded set H of real numbers into R such that
range f H of f fails to be bounded.
We can define f x 1 for 0 x 1.
x
5. Prove that if a set H of real numbers is unbounded above and f x x for every number x in H, then f is a
continuous function on H and f fails to have a maximum.
Since the range of f is the set H which is assumed to be unbounded, the function f must be
unbounded above.
6. Prove that if S is a subset of a metric space X and if a is a point of X that is close to S but not a member of S
then the function f defined by the equation
1
fx
d a, x
for every x X is unbounded.We can see that f has no maximum by showing that f is unbounded
above. Given any positive number q the inequality
fx q
says that
1
q
d a, x
which holds when
d a, x 1
q
But, since a is close to the set S, we know that there do indeed exist members x of S for which the
inequality 227 a 1
q
holds. Therefore there are members x in H for which f x q and we have shown that f fails to be
bounded above.
x 7. Is it true that if every continuous function from a given metric space X to 0, Ý has a minimum then X must
be compact?
Yes, it’s true. The theorems of this section tell us that if a metric space X is not compact then there
exists a continuous unbounded real function on X. For any such function f, the function
1
g
1 f2
is continuous (and bounded) on X and has no minimum.
8. Is it true that if X is a complete metric space and f is a continuous function from X onto a metric space Y then
Y must also be complete?
The answer is no. Define
fx 1
x
for every number x in the complete metric space 1, Ý . The range of f is the metric space 0, 1 ,
which is not complete.
9. Is it true that if X is a totally bounded metric space and f is a continuous function from X onto a metric space
Y, then Y must also be totally bounded?
Of course not. We have seen continuous functions on totally bounded spaces such as 0, 1 whose
ranges are not bounded.
10. Give an example of a metric space X and two closed subsets H and K of X such that H
fact that
inf d x, y
x H and y K 0. K in spite of the Take X R and
Z H n n 2 and
K n 1
n Z n 2 For another example, take X R and
2 H x, 1
x x 1 K x, 1
x x 1 and 11. Suppose that X is a metric space, that H and K are closed subsets of X and are disjoint from each other and
that the subspace K of X is compact. Prove that
x H and y K 0.
inf d x,...
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 Fall '08
 STAFF
 Math, Calculus

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