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12
SPRING 2012
5.
Partial Derivatives, continued
The extra condition on
f
that provides a converse for Proposition 4.5 is the continuity of
its partial derivatives, in the sense that
x
°→
D
j
f
i
(
x
)shou
ldbecon
t
inuou
sf
rom
U
⊆
R
n
into
R
for each
i, j
. (See Proposition 5.3.)
Defnition 5.1.
A function
f
:
U
⊆
R
n
→
R
m
is
continuously di
f
erentiable
if
f
is di
f
er-
entiable at each
x
∈
U
,andth
emap
x
°→
f
°
(
x
)f
rom
U
into
L
(
R
n
,
R
m
)i
scon
t
inuou
s
.
The set of continuously di
f
erentiable functions from
U
into
R
m
is denoted
C
1
(
U,
R
m
), and
the expression “
f
is
C
1
”isanabbrev
iat
ionforthestatemen
t
f
∈
C
1
(
U,
R
m
).
C
1
(
U,
R
)is
abbreviated to
C
1
(
U
).
Continuous di
f
erentiability is a stronger condition than di
f
erentiability, even for real-
valued functions of a single real variable.
Example 5.2.
DeFne
f
:
R
→
R
by
f
(
x
)=
°
x
2
sin
1
x
if
x
±
=0
0i
f
x
=0
.
Then
f
°
(
x
)ex
istsforeach
x
, but the function
x
°→
f
°
(
x
) is not continuous.
Proposition 5.3.
A function
f
:
U
⊆
R
n
→

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