Math 413, Fall semester, 2007
Bigtimes Homework set 6, handed out September 27, due October 4
Remember that there is a prelim scheduled for October 4, 7:30 pm. I don’t know the
room yet, but I will announce it as soon as I do.
The curriculum is the text through
Chapter 4.
You should have read everything through Chapter 4.
Here are some review exercises.
Problem 1.
Consider the function
f
:
R
→
R
defined by
f
(
x
) =
‰
1
q
if
x
∈
Q
and
x
=
p
q
with
p, q
coprime
0
if
x
is irrational.
Show that
f
is continuous at all irrationals.
Problem 2.
A function
f
: [
a, b
] is
convex
if for all
x, y
satisfyling
a
≤
x < y
≤
b
and all
t
∈
(0
,
1) we have
f
(
tx
+ (1

t
)
y
)
≤
tf
(
x
) + (1

t
)
f
(
y
)
.
(a) Show that every convex function is continuous.
(b) Show that if
f
is convex, then so is
e
f
.
Show that
f
: [
a, b
]
→
R
is convex if and only if it satisfies
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 HUBBARD
 Math, Derivative, Continuous function, Convex function

Click to edit the document details