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Unformatted text preview: an we say about the continuity of the function f g at the number a?
Were the function f g to be continuous at 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 number a?
If f a
0 then, since the equation
fxgx
gx
fx
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 number a if f a 0 and g is a bounded
function?
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  pf 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 S
a , a we have
f x g x faga  pf x  p p . d. What can be said about the continuity of the function fg if f a
0?
The purpose of this part of the question is to prod people who may not have done Part a.
11. Give an example of two functions f and g that are both discontinuous at a given number a such that their sum
f g is continuous at a.
We define
1 if x 2
fx 0 if x 2 1 1
gx if x 2 and if x 2 0 if x 2
1 if x 2 12. Given that f is a continuous function from a closed set H into R and that a H, prove that the set
E x H fx fa
is closed. Hint: Consider the behavior of a convergent sequence in the set E. 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
lim
f x n Ý f xn n Ý f a f a
lim
and so x E as we promised. 13. Given that f and g are continuous functions from R to R and that
E x R fx gx 194 prove that the set E must be closed.
This exercise is very similar to Exercise 12. 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
lim
f x n Ý f xn n Ý g xn g x
lim
and so x E as we promised. 14. 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.
15. 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...
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 Fall '08
 STAFF
 Math, Calculus

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