SelfAssessment 1 – Math 104A, Fall 2010
Answer the following questions without looking in the book. If you do not
feel comfortable answering them, read the corresponding sections in the book,
and then solve the problems, again without looking in the book.
1. A function
f
: [
a, b
]
→
R
is said to satisfy a
Lipschitz condition
with
Lipschitz constant
L
on [
a, b
] if, for every
x, y
∈
[
a, b
], we have

f
(
x
)
−
f
(
y
)
 ≤
L

x
−
y

.
(a) Show that if
f
satisfies a Lipschitz condition with Lipschitz con
stant
L
on [
a, b
], then
f
is continuous on [
a, b
].
(b) Show that if
f
has a derivative that is bounded on [
a, b
] by
L
,
then
f
satisfies a Lipschitz condition with Lipschitz constant
L
on
[
a, b
].
(c) Give an example of a function that is continuous on a closed in
terval but does not satisfy a Lipschitz condition on the interval.
2. Let
f
∈
C
([
a, b
]), and consider
p
∈
(
a, b
).
(a) Suppose
f
(
p
)
negationslash
= 0.
Show that there exists a
δ >
0 such that
f
(
x
)
negationslash
= 0 for all
x
∈
[
p
−
δ, p
+
δ
]
⊂
[
a, b
].
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 Fall '08
 Staff
 Math, lim, Continuous function, Metric space, pn, lipschitz condition

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