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for some point (x*,y*) on the line segment from (x
0
,y
0
) to (x,y).
Proof.
Let
p
= (x,y),
p
0
= (x
0
,y
0
), and define
()
=
(
(
)
)
Œ
[]
Ft
f
ppp
+
t
-
,
t
01
, .
0
0
The theorem follows easily from the chain rule and the basic Taylor polynomial for
functions of one variable applied to F (Theorem D.2.3). See [Buck78] .
So far in this section differentiability was a notion that was defined only for a
function f whose domain
A
was an open set; however, one can define the derivative
of a function also in cases where its domain is a more general set. Basically, all one
has to be able to do is extend the function f to a function F defined on an open set
containing
A
. One then defines the derivative of f to be the derivative of F and shows
that this value does not depend on the extension F one has chosen. In fact, one only
really needs local extensions, that is, for every point in
A
we need to be able to extend
f to a differentiable function on a neighborhood of that point.
Let
A
be an arbitrary subset of
R
n
. A map f :
A
Æ
R
m
Definition 1.
is said to be
of
class C
k
or a
C
k
map on
A
if there exists an open neighborhood
U
of
A
in
R
n
and a
C
k
map
m
F
:
UR
Æ
that extends f, that is, f = F | (
U
«
A
). If k ≥ 1, then the
rank of f at a point
p
is the
rank of DF at
p
.
Let
A
be an arbitrary subset of
R
n
. A map f :
A
Æ
R
m
Definition 2.
is said to be
of
class C
k
or a
C
k
map at a point
p
in
A
if there exists a neighborhood
U
p
of
p
in
R
n
and a C
k
map
m
F
p
UR
:
Æ
that extends f | (
U
p
«
A
). If k ≥ 1, then the
rank of f at the point
p
is the rank of DF at
p
. The map f is a
C
k
map
if it is of class C
k
at every point
p
in
A
.
4.3.25. Theorem.
The definitions of C
k
maps on a set or at a point are well defined.
The two definitions of C
k
maps on a set are equivalent. The notion of rank is well
defined in all cases. If the set
A
is open, then the definitions agree with the earlier
definition of differentiability and rank.
Proof.
For details see [Munk61].
Notice that neither definition actually defined a derivative although we did define
the rank of the map. Since the extensions F are not unique, it is not possible to define
a derivative in general. In certain common cases, such as rectangles or disks where
boundary points have nice “half-space” neighborhoods, the derivative
is
defined
uniquely. Actually, in such cases, one could simply define the derivative at such a point
