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Index
9.11
Theorem (Intermediate Value Theorem.)
Let
with , and let
be a
continuous function. Suppose . Then there is some point
with .
Proof: We will construct a binary search sequence with
such that

(9.12) 
Let
This is a binary search sequence satisfying condition (9.12).
Let be the number such that
.
Then and (cf theorem 7.87),
so by
continuity of ,
and
. Since for all , it follows by the inequality theorem that
, and since , we have
. Hence,
.
9.13
Exercise (Intermediate value theorem.)
A
Let
with
and let
be a continuous
function with
. Let
be a number in the interval
. Show that there is some
with
. (Use
theorem
9.11. Do not reprove it.)
9.14
Notation ( is between and .)
Let
. I say
is
between and
if either
or
.
9.15
Corollary (Intermediate value theorem.)
Let
with . Let
be a continuous function with
. If is any number between and , then there is some
such that . In particular, if and have opposite
signs, there is a number with .
Proof: By exercise 9.13A, the result holds when . If
, let . Then is continuous on and , so by
exercise 9.13A there is a with , so so
.
9.16
Example.
Let
be real numbers with
, and let
We will show that there is a number
such that
. Suppose, in order
to get a contradiction, that no such number
exists, and let
(I use the fact that
has no zeros here.) Then
It follows that
for some
, so
and
have opposite signs
for some
, and
is continuous on
, so by the intermediate value
theorem,
for some
, contradicting the assumption that
is never
zero.
9.18
Exercise.
Let
. Prove that the equation
has at least three
solutions in
.
s
9.19
Exercise.
A
Let
be a continuous function
from
to
such that
 a)
 For all
,
.
 b)
 .
Prove that
.
9.20
Note.
The intermediate value theorem was proved independently by
Bernhard Bolzano in
1817 [
42], and Augustin Cauchy in 1821[
23, pp 167168]. The
proof we have given is almost identical with Cauchy's proof.
The extreme value theorem was proved by Karl
Weierstrass circa 1861.
Next: 10. The Derivative
Up: 9. Properties of Continuous
Previous: 9.1 Extreme Values
Index