This preview shows pages 1–2. Sign up to view the full content.
Intro to Modern Analysis
Summer MMXI Final Exam
Name
DIES LVNAE AD VII KAL AVG MMDCCLXIV
*
Do all problems, in any order.
Explain all answers. An unju ified response alone may not receive full credit.
No notes, texts, or calculators may be used on this exam.
Problem
Possible
Points
Points
Earned
1
5
2
5
3
5
4
5
5
5
6
5
TOTAL
30
GOODLUCK
*
August 11, 2011
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 1. 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
.
2. Show that the inverse function theorem follows from the implicit function theorem.
3. Suppose that
U
⊆
R
n
is open and that
f
:
U
→
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 to
prove that
f
has a saddle point at
a
.
4. Recall that a point
x
∈
R
is a
condensation point of a set
A
if every neighborhood of
x
contains uncountably many points in
A
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 10/18/2011 for the course MATH S4062Q taught by Professor Staff during the Summer '11 term at Columbia.
 Summer '11
 Staff

Click to edit the document details