Rudin_TF03-anno
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Hong Kong Polytechnic University *
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Course
273
Subject
Mathematics
Date
Nov 24, 2024
Type
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4
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1
Chapter 5
Q1.
Say whether each of the following statements is true or false.
(a) If a bounded function
R
!
R
is continuous, then
f
0
(
t
) exists for all real numbers
t
.
(b) If
f
is a di
↵
erentiable real-valued function on [0
,
2] and lim
t
!
1
f
(
t
) = 7, then
f
(1) = 7.
(c) If
f
and
g
are real-valued functions on [
a, b
], then at any point where
f
and
g
are both
di
↵
erentiable,
f
+
g
is also di
↵
erentiable.
(d) If
f
and
g
are real-valued functions on [
a, b
], then at any point where
f
and
f
+
g
are
both di
↵
erentiable,
g
is also di
↵
erentiable.
Solution. (a) F.
(b) T.
(c) T.
(d) T.
Q2.
Say whether each of the following statements is true or false.
(a) The function
f
(
x, y
) =
x
2
-
y
2
on
R
2
has a local maximum at (0
,
0).
(b) If
f
:
R
!
R
is di
↵
erentiable, and
x
is a point such that
f
0
(
x
) = 0, then
f
has either a
local maximum or a local minimum at
x
.
(c) If
f
:
R
!
R
is everywhere di
↵
erentiable, and satisfies
f
(
n
) =
n
for all integers
n
, then
there are infinitely many real numbers
x
such that
f
0
(
x
) = 1.
Solution. (a) F.
(b) F.
(c) T.
Q3.
Say whether each of the following statements is true or false.
(a) If
f
:
R
!
R
is di
↵
erentiable, and
f
0
(
x
) is
>
10 for all negative
x
and
<
10 for all
positive
x
, then
f
0
(0) = 10.
(b) If
f
: (0
,
1)
[
(2
,
3)
[
(4
,
5)
!
R
is di
↵
erentiable, and
f
0
is negative for all
x
2
(0
,
1) and
positive for all
x
2
(4
,
5), then it must be zero for some
x
2
(2
,
3).
Solution. (a) T.
(b) F.
1
Q4.
Say whether each of the following statements is true or false.
(a) If
f
:
R
!
R
is a di
↵
erentiable function satisfying
f
0
(
x
) =
e
x
2
, then
lim
x
!
+
1
f
(
x
)
/f
0
(
x
) =
0.
(b) If
f
:
R
!
R
is a di
↵
erentiable function satisfying
f
0
(
x
) =
e
-
x
2
, then
lim
x
!
+
1
f
(
x
)
/f
0
(
x
) =
0.
Solution. (a) T.
(b) F.
Q5.
Say whether each of the following statements is true or false.
(a) If
f
:
R
!
R
is a function such that
f
(
n
)
exists for all positive integers
n
, then
f
(
x
) =
1
X
n
=0
⇣
f
(
n
)
(0)
x
n
⌘
/n
! for all real numbers
x
such that this series converges.
(b) If
f
:
R
!
R
is a function such that
f
(
n
)
(
x
) exists and is
1
,
000 for all positive integers
n
and all
x
2
R
, then
f
(
x
) =
1
X
n
=0
⇣
f
(
n
)
(0)
x
n
⌘
/n
! for all
x
2
R
.
Solution. (a) F.
(b) T.
2
2
Chapter 6
Q1.
Say whether each of the following statements is true or false.
(a) In the equation
U
(
P, f,
↵
) =
n
X
i
=1
M
i
Δ
↵
i
(used in Rudin’s definition of the Riemann-
Stieltjes integral),
M
i
denotes
f
(
x
i
).
(b) In the same equation,
n
denotes the number of intervals [
x
i
-
1
, x
i
] into which the partition
P
divides [
a, b
].
(c) If a partition
P
⇤
contains more points than a partition
P
, then
P
⇤
is a refinement of
P
.
(d) If
P
⇤
is a refinement of the partition
P
of the interval [
a, b
], then for every bounded
function
f
and increasing function
↵
on [
a, b
],
U
(
P
⇤
, f,
↵
)
-
L
(
P
⇤
, f,
↵
)
U
(
P, f,
↵
)
-
L
(
P, f,
↵
).
(e) If
f
is a bounded function and
↵
an increasing function on [
a, b
], and if there ex-
ist partitions
P
1
,
P
2
, . . .
of [
a, b
] such that for each
i
,
U
(
P
i
+1
, f,
↵
)
-
L
(
P
i
+1
, f,
↵
)
1
2
(
U
(
P
i
, f,
↵
)
-
L
(
P
i
, f,
↵
)), then
f
2
R
(
↵
).
(f) If
f
is a bounded function and
↵
an increasing function on [
a, b
], and if
P
1
✓
P
2
✓
. . .
are
a sequence of partitions, each a refinement of the one before, then
inf
n
=1
,
2
,...
(
U
(
P
n
, f,
↵
)) =
Z
fd
↵
.
Solution. (a) F.
(b) T.
(c) F.
(d) T.
(e) T.
(f) F.
Q2.
Say whether each of the following statements is true or false.
(a) If
|
f
(
x
)
|
|
g
(
x
)
|
for all
x
2
[
a, b
], and
g
2
R
(
↵
), then
f
2
R
(
↵
).
Solution. (a) F.
Q3.
Say whether each of the following statements is true or false.
(a) If
f
2
R
(
↵
) on an interval [
a, b
], then
f
2
R
(
↵
) on every subinterval [
c, d
]
✓
[
a, b
].
Solution. (a) T.
3
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Q4.
Say whether each of the following statements is true or false.
(a) There exists an increasing function
↵
on [0
,
1] such that for every continuous function
f
on that interval,
Z
1
0
fd
↵
=
X
2
-
n
f
(1
/n
).
(b) There exists an increasing function
↵
on [0
,
1] such that for every continuous function
f
on that interval,
Z
1
0
fd
↵
=
X
(1
/n
)
f
(
2
-
n
)
(c) If
↵
increases monotonically on [
a, b
] and is di
↵
erentiable, then
↵
0
2
R
on [
a, b
].
(d) If
f
2
R
on [0
,
1], then
Z
1
0
f
(
x
)
dx
=
Z
1
/
2
0
f
(2
x
)
d
(2
x
).
Solution. (a) T.
(b) F.
(c) F.
(d) T.
Q5.
Say whether each of the following statements is true or false.
(a) If
f
2
R
on [
a, b
], and for all
x
2
[
a, b
] we define
F
(
x
) =
R
x
a
f
(
t
)
dt
, then
F
is di
↵
erentiable
and
F
0
=
f
.
(b) If
f
is a continuous function on [
a, b
], and for all
x
2
[
a, b
] we define
F
(
x
) =
R
b
x
f
(
t
)
dt
,
then
F
is di
↵
erentiable and
F
0
=
-
f
.
Solution. (a) F.
(b) T.
4