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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
![\begin{displaymath}
f(a_n)\leq 0\leq f(b_n)\mbox{ for all } n.
\end{displaymath}](img974.gif) |
(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
![$f(c)=\lim\{f(a_n)\}$](img981.gif)
. Hence,
.
9.13
Exercise (Intermediate value theorem.)
A
Let
![$a,b\in\mbox{{\bf R}}$](img395.gif)
with
![$a<b$](img933.gif)
and let
![$f:[a,b]\to\mbox{{\bf R}}$](img983.gif)
be a continuous
function with
![$f(a)<f(b)$](img984.gif)
. Let
![$y$](img785.gif)
be a number in the interval
![$\left(f(a),f(b)\right)$](img985.gif)
. Show that there is some
![$c\in(a,b)$](img972.gif)
with
![$f(c)=y$](img986.gif)
. (Use
theorem
9.11. Do not reprove it.)
9.14
Notation (
is between
and
.)
Let
![$a,b,x\in\mbox{{\bf R}}$](img987.gif)
. I say
![$x$](img428.gif)
is
between ![$a$](img590.gif)
and
![$b$](img954.gif)
if either
![$a<x<b$](img988.gif)
or
![$b<x<a$](img989.gif)
.
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
![$A,B,C,D$](img678.gif)
be real numbers with
![$A\neq 0$](img997.gif)
, and let
We will show that there is a number
![$c\in\mbox{{\bf R}}$](img494.gif)
such that
![$f(c)=0$](img973.gif)
. Suppose, in order
to get a contradiction, that no such number
![$c$](img528.gif)
exists, and let
(I use the fact that
![$f(x)$](img1000.gif)
has no zeros here.) Then
It follows that
![$g(n)<0$](img1002.gif)
for some
![$n$](img313.gif)
, so
![$f(-n)$](img1003.gif)
and
![$f(n)$](img304.gif)
have opposite signs
for some
![$n$](img313.gif)
, and
![$g$](img111.gif)
is continuous on
![$[-n,n]$](img1004.gif)
, so by the intermediate value
theorem,
![$g(c)=0$](img995.gif)
for some
![$c\in(-n,n)$](img1005.gif)
, contradicting the assumption that
![$g$](img111.gif)
is never
zero.
9.18
Exercise.
Let
![$f(x)=x^3-3x+1$](img1008.gif)
. Prove that the equation
![$f(x)=0$](img1009.gif)
has at least three
solutions in
![$\mbox{{\bf R}}$](img4.gif)
.
s
9.19
Exercise.
A
Let
![$F$](img733.gif)
be a continuous function
from
![$\mbox{{\bf R}}$](img4.gif)
to
![$\mbox{{\bf R}}$](img4.gif)
such that
- a)
- For all
,
.
- b)
.
Prove that
![$F(4)>0$](img1012.gif)
.
9.20
Note.
The intermediate value theorem was proved independently by
Bernhard Bolzano in
1817 [
42], and Augustin Cauchy in 1821[
23, pp 167-168]. 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
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