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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
0i
f
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
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 Fall '07
 HUBBARD
 Math

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