Practice Midterm II
1. Show that the function
f
(
x
) =
(
x
sin(

1
/x
)
x
6
= 0
0
x
= 0
is continuous but not differentiable at 0.
2. Find
all
the values of
x
for which the series
∞
X
k
=1
x
k
√
k
converges.
3. Suppose that
f
: (
a, b
)
→
R
is differentiable on (
a, b
) and
f
0
(
x
) is bounded on
(
a, b
). Prove that
f
is uniformly continuous on (
a, b
).
4. Let
f
be a continuous function mapping the interval [
a, b
] into the interval [
a, b
]
(in other words,
f
: [
a, b
]
→
[
a, b
]). Show that there is a point
z
in [
a, b
] such that
f
(
z
) =
z
(that is, show that
f
has a fixed point).
5. Give an example of a
divergent
series
∑
∞
k
=1
a
k
whose partial sums are bounded.
1
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2
Solutions
1. Since sin(
x
) is bounded by 1, we have

x
sin(1
/x
)
 ≤ 
x
 →
0 as
x
→
0
,
which shows that
f
(
x
) is continuous at 0. Now
lim
x
→
0
f
(
x
)

f
(0)
x
= lim
x
→
0
sin(1
/x
)
does not exist; we can show that by letting
a
k
=
2
πk
and observing that
sin(1
/a
k
) = sin(
π
2
k
) =
(
0
for even k
±
1
for odd k
.
It follows, of course, that
f
(
x
) is not differentiable at 0.
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 Winter '10
 Bremer
 Calculus, lim, Continuous function

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