Modern Analysis, Homework 5
1. We saw that the implicit function theorem follows from the inverse function theorem.
Show that the inverse function theorem follows from the implicit function theorem.
2. Let
A,B
⊆
R
n
. Set
d
(
A,B
) = inf
{
x

y

;
x
∈
A,y
∈
B
}
.
If
A
is compact and
B
is closed, prove that
d
(
A,B
)
>
0 if
A
∩
B
=
{}
.
3. Prove that
A
⊆
R
n
is measurable if and only if
A
=
G
\
Z
for some
G
δ
set
1
and some
set
Z
of measure 0.
4. Suppose that
f
:
U
⊆
R
n
→
R
has continuous secondorder partial derivatives on some
ball
B
r
(
a
) for some
a
∈
U
, and that
∇
f
(
a
) =
~
0. Suppose that the Hessian of
f
at
a
has both positive and negative eigenvalues. Use Taylor’s theorem in several variables
to prove that
f
has a saddle point at
a
.
2
5. (a) Let
E
⊂
R
be measurable with
μ
(
E
)
>
0. Show that the set of diﬀerences
{
d
=
x

y, x
∈
E,y
∈
E
}
contains an interval centered at the origin. (
Hint:
Find
an open set
G
containing
E
such that
μ
(
G
)
<
4
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 Summer '11
 Staff
 measure, implicit function theorem, Inverse Function Theorem, Lebesgue measure, Nonmeasurable set

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