This preview shows page 1. Sign up to view the full content.
3.6. EXTREMA
319
Theorem 3.6.6
(Extreme Value Theorem)
.
If
S
⊂
R
3
is compact, then
every realvalued function
f
which is continuous on
S
achieves its
minimum and maximum on
S
.
Note that this result includes the Extreme Value Theorem for functions of
one variable, since closed intervals are compact, but even in the single
variable setting, it applies to functions continuous on sets more general
than intervals.
Proof.
The strategy of this proof is: Frst, we show that
f
must be
bounded on
S
, and second, we prove that there exists a point
s
∈
S
where
f
(
s
) = sup
x
∈
S
f
(
x
) (
resp
.
f
(
s
) = inf
x
∈
S
f
(
x
)).
12
Step 1:
f
(
x
)
is bounded on
S
:
Suppose
f
(
x
) is
not
bounded on
S
: this
means that there exist points in
S
at which

f
(
x
)

is arbitrarily high: thus
we can pick a sequence
s
k
∈
S
with

f
(
s
k
)

> k
. Since
S
is (sequentially)
compact, we can Fnd a subsequence—which without loss of generality can
be assumed to be the whole sequence—which converges to a point of
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus

Click to edit the document details