Remark:
The defnition oF regular point and critical point in de la ±uente (as well as in Mas-Colell,
Whinston, and Green) is di²erent: they use
m
rather than min
{
m, n
}
. I think the defnition I have given
is more natural. IF
m
≤
n
, the two are equivalent. IF
m>n
, then since Rank (
df
x
)
≤
min
{
m, n
}
,then
every
x
∈
X
will be a critical point in the de la ±uente and MWG defnitions, and every
y
∈
f
(
X
) will be a
critical value. In the defnition I have given, a point is critical iF the rank is smaller than the largest it could
possibly be. The two important theorems (Sard’s Theorem and the Transversality Theorem) concerning
critical values are true with either defnition.
Example:
Consider the Function
f
:(0
,
2
π
)
→
R
defned by
f
(
x
)=s
in
x
Then
f
0
(
x
)=co
s
x
,so
f
0
(
x
)=0Fo
r
x
=
π/
2and
x
=3
2.
Df
(
x
)i
sth
e1
×
1 matrix (
f
0
(
x
)), so
Rank
x
=Rank
Df
(
x
) = 1 iF and only iF
f
0
(
x
)
6
= 0. Thus, the critical points oF
f
are
2and3
2, so
the set oF regular points oF
f
is
(0
,π/
2)
∪
(
2
,
3
2)
∪
(3
2
,
2
π
)
The critical values oF
f
are
f
(
2) = sin(
2) = 1 and
f
(3
2) = sin(3
2) =
−
1; the set oF regular values
oF
f
is
(
−∞
,
−
1)
∪
(
−
1
,
1)
∪
(1
,
∞
)
Notice that 0 is not a critical value. Given
α
∈
R
, consider the perturbed Function
f
α
(
x
)=
f
(
x
)+
α
Notice that
f
0
α
(
x
f
0
(
x
), so the critical points oF
f
α
are the same as those oF
f
.±
o
r
α
close to zero, the
solution to the equation
f
α
(
x
)=0
2