Math 312, Intro. to Real Analysis:
Homework #6 Solutions
Stephen G. Simpson
Friday, April 10, 2009
The assignment consists of Exercises 17.3(a,b,c,f), 17.4, 17.9(c,d), 17.10(a,b),
17.14, 18.5, 18.7, 19.1, 19.2(b,c), 19.5 in the Ross textbook. Each exercise counts
10 points.
17.3.
(a) By Theorem 17.4(ii) applied three times, cos
4
x
is continuous. Then,
by Theorem 17.4(i), 1 + cos
4
x
is continuous.
Then, by Theorem
17.5, log
e
(1 + cos
4
x
) is continuous. Since 1 + cos
4
x
≥
1 for all
x
, the
domain of this function is the entire real line.
(b) Since
x
π
=
e
π
log
x
, it is clear by Theorem 17.4 that
x
π
is continuous.
It then follows as usual that (sin
2
x
+ cos
6
x
)
π
is continuous. Again,
the domain of this function is the entire real line.
(c) We have 2
x
2
=
e
x
2
log2
so as before this is continuous.
Again, the
domain is the entire real line.
(f) By Theorem 17.4(iii), 1
/x
is continuous for all
x
negationslash
= 0. It then follows
as usual that
x
sin(1
/x
) is continuous for all
x
negationslash
= 0.
17.4. Example 5 in
§
8 says that if lim
x
n
=
x
and
x
n
≥
0 for all
n
, then
lim
√
x
n
=
√
x
. According to Definition 17.1 this says precisely that
√
x
is continuous for all
x
≥
0.
17.9.
(c) Let
f
(
x
) =
x
sin(1
/x
) for
x
negationslash
= 0, and let
f
(0) = 0. We wish to show
that
f
is continuous at 0 using the
ǫ

δ
definition of continuity. Given
ǫ >
0, we must find
δ >
0 such that

f
(
x
)

< ǫ
whenever

x

< δ
. We
have

f
(
x
)

=

x
 · 
sin(1
/x
)
 ≤ 
x

since

sin
y
 ≤
1 for all
y
. So let
δ
=
ǫ
. Then clearly

x

< δ
implies

x

< ǫ
which implies

f
(
x
)

< ǫ
,
Q.E.D.
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 Spring '09
 Math, Continuous function, entire real line

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