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.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 ALL
 Calculus, Intermediate Value Theorem, Continuous function

Click to edit the document details