MATH 3220 § 1
ELEVENTH HOMEWORK ASSIGNMENT
Nov. 28, 2000
A. Treibergs
Due Dec. 5, 2000
Please hand in the following exercises from the text "Introduction to Analysis, Second Edition" by William Wade, Prentice Hall 1999.
380[ 2, 4a, 7ab, 10 ].
Here are the equivalent exercises for those using the First Edition:
346[ 4a, 10 ]
Prove that every finite subset of
R
^{n}
is a Jordan region of volume zero.
Show that even in
R
^{2}
, part a. is not true if "finite" is replaced by "countable."
By an interval in
R
^{2}
we mean a set of the form
for some real a, b, c. Prove that every interval in
R
^{2}
is a Jordan region.
Suppose that V is a bounded open set of
R
^{n}
and g:V ->
R
^{n}
is continuously differentiable on V.
Let H be a compact subset of V. Show that there is a constant C>0 (depending only on g, H, n) so that for all cubes Q in H,
___
Show that if E, a subset of V has volume zero then g(E) has volume zero.