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218
CHAPTER 3. REALVALUED FUNCTIONS: DIFFERENTIATION
Remark
3.1.2
can also be used to get a weak analogue of the Intermediate
Value Theorem (
Calculus Deconstructed
, Theorem 3.2.1). Recall that this
says, for
f
:
R
→
R
continuous on [
a,b
], that if
f
(
a
) =
A
and
f
(
b
) =
B
then
for every
C
between
A
and
B
the equation
f
(
x
) =
C
has at least one
solution between
a
and
b
. Since the notion of a point in the plane or in
space being “between” two others doesn’t really make sense, there isn’t
really a direct analogue of the Intermediate Value Theorem, either for
−→
f
:
R
→
R
3
or for
f
:
R
3
→
R
. However, we can do the following: Given two
points
−→
a ,
−→
b
∈
R
3
, we deFne a
path
from
−→
a
to
−→
b
to be the image of any
locally onetoone continuous function
−→
p
:
R
→
R
3
, parametrized so that
−→
p
(
a
) =
−→
a
and
−→
p
(
b
) =
−→
b
. Then we can talk about points “between”
−→
a
and
−→
b
along this curve
.
Proposition 3.1.4.
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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

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