Math 301/ENAS 513: Homework 10 Solution
Triet M. Le
Page 104: 2:
f
is real on
R
and

f
(
x
)

< M
for all
x
∈
R
. Then for any
x
=
y
, the
mean value theorem implies
f
(
x
)

f
(
y
)
x

y
=
f
(
c
)
,
for some
c
between
x
and
y.
Hence, for any
>
0,

f
(
x
)

f
(
y
)

=

f
(
c
)

x

y
 ≤
M

x

y

<
,
whenever,

x

y

< δ
=
/M
. Thus,
f
is uniformly continuous on
R
.
Page 104: 3a:
Fix
>
0. Since
f
(
x
)
→
0 as

x
 → ∞
, there exists
N >
0 such that

f
(
x
)

<
/
2
,
if
x /
∈
[

N, N
]
.
f
is obviously continuous on [

(
N
+ 1)
, N
+ 1] which is compact, hence it is uniformly
continuous. This implies there exists 0
< δ <
1 such that

f
(
x
)

f
(
y
)

<
,
whenever
x, y
∈
[

(
N
+ 1)
, N
+ 1] and

x

y

< δ.
Now, for any
x, y
∈
R
such that

x

y

< δ
. if both
x, y
∈
[

(
N
+ 1)
, N
+ 1], then by the
above equation,

f
(
x
)

f
(
y
)

<
.
On the otherhand if either
x /
∈
[

(
N
+1)
, N
+1] and/or
y /
∈
[

(
N
+1)
, N
+1], the condition

x

y

< δ <
1 implies that both
x, y /
∈
[

N, N
]. This implies,

f
(
x
)

f
(
y
)
 ≤ 
f
(
x
)

+

f
(
y
)

<
.
Thus, in all cases,

f
(
x
)

f
(
y
)

<
whenever

x

y

< δ
, which shows that
f
is uniformly
continuous on
R
.
Page 104: 3b:
Consider
f
(
x
) = sin(
x
4
)
/
(
x
2
+ 1). Clearly,
f
is continuous and differen
tiable on
R