Unformatted text preview: a, it would follow from the fact that
g fg f
that g is continuous at a. Therefore f g cannot be continuous at a.
b. What can we say about the continuity of the function fg at the point a?
If f a
0 then, since the equation
would hold for every number x sufficiently close to a, we could use an argument like the one we
used in Part a to deduce that the function fg cannot be continuous at a.
However, execise 2 shows us to functions, one continuous at 0 and the other discontinuous at
0 whose product is continuous at 0.
c. What can we say about the continuity of the function fg at the point a if f a 0 and g is a bounded
Choose a number p such that |g x | p for each x S. Whenever x S we have
|f x g x f a g a | |f x g x | p|f x |
and we can now use this inequality to prove that fg must be continuous at a.
Suppose that 0. Using the fact that f is continuous at w and the fact that /p 0 we choose
a number 0 such that the condition |f x | p whenever x S
a , a . Then,
whenever x B a, we have
|f x g x faga | p|f x | p p . d. What can be said about the continuity of the function fg if f a
The purpose of this part of the question is to prod people who may not have done Part a.
16. Give an example of two functions f and g that are both discontinuous at a given point a such that their sum
f g is continuous at a.
1 if x 2
fx 0 if x 2 1 1 if x 2 and gx if x 2 0 if x 2
1 if x 2 17.
a. Given that f is a continuous function from a closed subset H of R into R and that a
set 223 H, prove that the E x H fx fa is closed. Solution: Suppose that x n is any convergent sequence in the set E. We shall show that the limit of
this sequence, that we shall call x, must also belong to E. Since the set H is closed we know that x H and
therefore we know that f is continuous at the number x. Therefore
f x n Ý f xn n Ý f a f a
and so x E as we promised. b. Given that f and g are continuous functions from a metric space X to a metric space Y and that
E x X fx gx
prove that the set E must be closed in X.
This exercise is very similar to part a. Suppose that x n is any convergent sequence in the set
E. We shall show that the limit of this sequence, that we shall call x, must also belong to E. We
now observe that
f x n Ý f xn n Ý g xn g x
and so x E as we promised. 18.
a. Given that f and g are continuous functions from R to R and that f x g x for every rational number x,
prove that f g.
The set of numbers x for which f x g x is closed and includes the set Q of rational numbers.
Therefore this set is all of R.
b. Given that f : Z R, that f 1 1 and that the equation
f xt f x f t
holds for all positive integers x and t, prove that f x x for every positive integer x. Solution: If you are familiar with the method of proof by mathematical induction which is
available in this book in an optional section then you should use mathematical induction to do this
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