Unformatted text preview: will follow at once from the obvious fact that f x 0 for
every number x. Exercises on the Distance Function
1. Two sets A and B of real numbers are said to be separated from each other if
A BA B .
Prove that if two sets A and B are separated from each other then
A x B x 0
whenever x A and
A x B x 0
whenever x B.
2. Prove that if two sets A and B are separated from each other then there exist two open sets U and V that are
disjoint from each other such that A U and B V. Solution: We define 197 U x R A x B x 0 V x R A x B x 0 . and
Since the function A B is continuous on R we deduce from the first of some earlier exercises that the
sets U and V are open.
3. Given two sets A and B of real numbers, prove that the following conditions are equivalent:
a. We have
b. There exists a continuous function f : R B . 0, 1 such that
fx 0 if x A 1 if x B . We deduce at once from Urysohn’s lemma that condition a implies condition b. On the other
hand, if f is a continuous function that takes the value 0 at every number x A and takes the
value 1 at every number x B then
4. Suppose that A, B and C are closed sets of real numbers and no two of these three sets intersect. Prove that
there exists a continuous function f on R such that
1 if x
2 if x B. 3 if x fx A C Using Urysohn’s lemma we choose two continuous functions g 1 and g 2 from R into 0, 1 such that
g 1 x 0 whenever x A and g 1 x 1 whenever x B Þ C and g 2 x 0 whenever x A Þ B and
g 2 x 1 whenever x C. We define g 3 x 1 for every number x. We now define
f g1 g2 g3.
5. Suppose that S is a set of real numbers and that S fails to be closed. Prove that there exists a convergent
sequence x n in S and a continuous function f on the set S such that the sequence f x n fails to converge. Solution: We begin by choosing a number w S S and we choose a sequence x n in S that
converges to the number w. We define E to be the range of the sequence x n and, Using the fact that that
the set E must be infinite, we choose an infinite subset A of E such that the set B E A is also infinite.
Since no member of S can lie in the closures of both A and B, the function
is continuous on the set S. Furthermore, since there must be infinitely many integers n for which x n A
and infinitely many integers n for which x n B the sequence f x n has no limit. Some Exercises on Continuous Functions on Closed
1. Give an example of a function f that is continuous on a closed set H such that the range f H of the function f 198 fails to be closed.
We can define f x 1 for x
is 0, 1 which is not closed. 1. This function f is continuous on the closed set 1, Ý and its range 2. Give an example of a function f that is continuous on a closed set H such that the range f H of the function 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 on a bounded set H such that the range f H of t...
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