Exercise 3c : Continuity

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Oct 10, 2013 (3 years and 10 months ago)

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Exercise 3c : Continuity

(1) Decide which of the following statements are true and which are false.

Prove the true ones and
provide counterexamples for the false ones.


(a) If

is continuous on

and
, then

is a closed,

bounded interval.



T h e s t a t e me n t i s t r u e;

B y t h e E x t r e me V a l u e T h e o r e m,

a t t a i n s i t

s s u p r e mu m a n d

i n f i mu m; t h a t i s, t h e r e a r e


s u c h t h a t

f o r a l l
. T h u s,

.

C o n v e r s e l y
, b y t h e I n t e r me d i a t e Va l u e T h e o r e m, f o r a n y

, t h e r e i s a n

( He r e w e ma y
a s s u me

)

s u c h t h a t
, h e n c e

.

T h u s,

.

T h e r e f o r e
,
w e c a n

c o n c l u d e t h a t
,

wh
ich is

indeed a closed interval.



(b) If

and

are continuous on
, if

and
,

then there is a

such that
.


The statement is true;


C o n s i d e r t h e f u n c t i o n

d e f i n e d b y
. C l e a r l y,

i s

c o n t i n u o u s o n
,
a n d
. B y t h e I n t e r me d i a t e V a l u e T h e o r e m, t h e r e i s a


s u c h t h a t
.
T h a t i s
,

o r
.



( c ) S u p p o s e t h a t

a n d

a r e d e
fi
ned and
fi
nite valued on an open interval


which contains
, that

is continuous at
, and that
. Then


is continuous at

i
ff


is
continuous at
.


The statement is true;


We

may
assume
. (If
, then consider
)

If

is continuous at
, then


is continuous at

by Theorem 3.22.

Conversely, suppose that

is continuous at
.

Since

is continuous at

and

, there is an open interval
,
w
here

, such that

is always
positive on
(See Lemma 3.28)
. In particular,

is nonzero on
.
Thus, we can consider the function
, which is clearly continuous at
.
Again, by Theorem 3.22,

is also continuous at
.




(d) Suppose that

and

are de
fi
ned and
fi
nite valued on
. If

and


are
continuous on
, then

is continuous on
.




T h e s t a t e me n t i s f a l s e;
T a k e
, w h i c h i s c o n t i n u o u s o n
, a n d

. T h e n
f o r a n y

, h e n c e

is

co
ntinuous on
, but

is discontinuous at
.




(2) Use limit theorems to show that the following functions are continuous

on
.

(
a
)



T h e a s s e r t i o n f o l l o w s f r o m T h e o r e m 3.2 2, 3.2 4, a n d t h a t

a r e
c o n t i n u o u s o n
.



(
b
)



B y T h e o r e m 3.2 2,


i s c o n t i n u o u s o n
.

, s o

i s c o n t i n u o u s a t
.

T h e r e f o r e,

i s c o n t i n u o u s o n
.



(
c
)



B y T h e o r e m 3.2 2

a n d 3.2 4
,

i s c o n t i n u o u s o n
.

s i n c e
, a n d
. S
o

i s c o n t i n u o u s a t
.

T h e r e f o r e,

i s c o n t i n u o u s o n
.




(
d
)




B y T h e o r e m 3.2 2

a n d 3.2 4
,

i s c o n
t i n u o u s o n
.

s i n c e

a n d

i s b o u n d e d

o n
. S
o

i s c o n t i n u o u s a t
.

Th
erefore,

is
continuous on
.




(3) For each of the following, prove that there is at least one

which

satis
fi
es the given
equation.

(
a
)





L e t

, t h e n

a n d


.
B y t h e I n t e r me d i a t e Va l u e T h e o r e m, t h e r e i s s o me

s u c h


t h a t
.



(
b
)




L e t
, t h e n

a n d


.
B y t h e I n t e r me d i a t e Va l u e T h e o r e m, t h e r e i s s o me

s u c h t h a t
.



(
c
)



L e t
, t h e n

a n d

.
B y t h e I n t e r me d i a t e Va l u e
T h e o r e m, t h e r e i s s o me

s u c h t h a t
.





(4) If

is continuous, prove that

is
fi
nite.



S i n c e

i s c o n t i n u o u s

o n

a n d t h e f u n c t i o n

i s c o n t i n u o u s o n
,

i s a l s o c o n t i n u o u s o n

b y T h e o r e m 3.2 4.
S o, b
y t h e E x t r e me V a l u e T h e o r e m
,


a t t a i n s i t

s s u p r e mu m o n
;

t h a t i s, t h e r e i s s o me

s u c h t h a t
, w h i c h i s f i n i t e
.




( 5) I f

i s c o n t i n u o u s, t h e n

h a s a
fi
xed

point
; that is, there

is a

such
that
.



L e t

b e d e f i n e d b y
. S i n c e

ma p s

i n t o
,

f o r a l l
.
T h u s,


a n d
.
I f

o r

, t h e n w e a r e d o n e. S u p p o s e t h a t

a n d
, t h e n b y t h e
I n t e r me d i a t e V a l u e T h e o r e m, t h e r e i s s o me

s u c h t h a t

o r
.




( 6) I f

i s a r e a l f u n c t i o n w h i c h i s c o n t i n u o u s a t

a n d i f

f o r

s o me
, p r o v e
t h a t t h e r e i s a n o p e n i n t e r v a l

c o n t a i n i n g

s u c h

t h a t

f o r a l l
.



S i n c e

i s c o n t i n u o u s a t

a n d
, t h e r e i s s o me

s u c h t h a t

i mp l i e s
. D
e f i n e
, t h e n f o r a n y
,

o r
.











(7) Show that there exist nowhere continuous functions

and

whose sum


is continuous
on
. Show that the same is true for the product of

functions.



L e t

a n d
, t h e n

a n d

a r e b o t h n o w h e r e
c o n t i n u o u s ( S e e E x a mp l e 3.3 2 ).

Ho w e v e r,

a n d

a r e
b o t h c o n t i n u o u s o n
.



( 8 ) S u p p o s e t h a t
, t h a t

i s a n o p e n i n t e r v a l c o n t a i n i n g
, t h a t
, a n d t h a t

i s
c o n t i n u o u s a t
. P r o v e t h a t

i s c o n t i n u o u s a t

i
f a n d o n l y i f

i s c o n t i n u o u s a t
.



Hi n t: A p p l y T h e o r e m 3.2 2, a n d n o t e t h a t
.



( 9) S u p p o s e t h a t

s a t i s
fi
es

for each
.

(a) Show that

for all

and
.



Hi n t:

U s e i n d u c t i o n t o p r o v e t h a t

f o r a l l

a n d
.




, h e n c e
.



, h e n c e
, a n d s o
.


( b ) P r o v e t h a t

f o r a l l

a n d
.




Hi n t:
U s e ( a ) a n d n o t e t h a t
.


( c ) P r o v e t h a t

i s c o n t i n u o u s a t

i
f a n d o n l y i f


i s c o n t i n u o u s o n
.




Hi n t:

i mp l i e s
.


( d ) P r o v e t h a t i f

i s c o n t i n u o u s a t
, t h e n t h e r e i s a n

s u c h t h a t


f o r a l l
.



We w i l l s h o w t h a t
. I f
, t h e n w e a r e d o n e.

I f
, t a k e a s e q u e n c e

i n

s u c h t h a t
. B y ( c ),
.


(10) Suppose that

satis
fi
es
. Modifying

the outline in the
above

Exercise,
show that if

is continuous at
, then

there is an

such that

for all
. (You may

assume that the function

is continuous on
.)





for all

and
.

Hint: Use induction to prove the case in which
;
, hence
;
, hence
.





for all

and
.

Hint:





is continuous at

i
f and only if


is continuous on
.

Hint:

implies
.

Note that


is always positive, hence

implies




I
f

is continuous at
, then there is an

such that


for all
.

We claim that
. If
, then we are done

by

.
If

,
then
take a sequence

in

such that
. By

,

is continuous on
, and since
the function

is
continuous on
,
we obtain

.




(11) If

is continuous and

,

prove that

has a minimum on
; that is, there is an

such that

.



Hi n t:
(
S e e t h e F i g u r e
)


U s e t h e a s s u mp t i o n


t o f i n d


a n d
.


U s e t h e a s s u mp t i o n
t h a t

i s c o n t i n u o u s


a
nd

the Extreme Value Theorem to find

.


Prove that
.


(12) Let
.

Assume that

and

for all
, and

that

for all

which satisfy
. For each
, de
fi
ne

.

(a) Prove that

exists and is
fi
nite for all
, and that

for all
. Thus

extends the
power of


function

from

to
.



L e t
.

F i r s t o f a l l, w e mu s t s h o w t h a t

i s n o n e mp t y.
T h i s
f o l l o w s f r o m
t h e A r c h i me d e a n P r o p e r t y w h i c h g u a r a n t e e s t h a t f o r a n y
, w e c a n f i n d a
r a t i o n a l n u mb e r


s ma l l e r t h a n
. S i mi l a r l y, w e c a n a l s o f i n d s o me

w i t h
,
w h i c h i mp l i e s

f o r a l l
; t h a t i s,

i s b o u n d e d
a b o v e
b y
.

B y t h e
C o mp l e t e n e s s A x i o m,

e x i s t s a n d i s

f i n i t e.
F i n a l l y, s i n c e

i s i t s e l f a n
u p p e r b o u n d f o r
, b y E x e r c i s e 1.3.7 ( a ),
.



( b ) I f

w i t h
, p r o v e t h a t
.



S i n c e
, b y t h e d e n s i t y o f

i n
, t h e r e i s s o me

w i t h
.
T h e n f o r a n y
, w h
e r e
, w e h a v e
,

h
e n c e
, a n d s o

i s a n
u p p e r b o u n d f o r
. T h u s,
. B y t h e d e f i n i t i o n o f
,
. F i n a l l y,

w i t h

i mp l i e s
. C o mb i n e t h e s e i n e q u a l i t i e s, w e o b t a i n
, w h i c h
c o mp l e t e s t h e p r o o f.




( c ) P r o v e t h a t t h e f u n c t i o n

i s c o n t i n u o u s o n
.



L e t

a n d

b e f i x e d
. B y E x e r c i s e 2.2 1,
, s o

t h e r e i s s o me

s u c h t h a t

i mp l i e s
. Ta k e
. T h e n f o r a n y

w i t h
, t h a t i s,
. Ta k e

w i t h
, t h e n
.

T h i s c o mp l e t e s t h e p r o o f.


(d) Prove that
,
, and

for all
.



L e t
, a n d

a n d

b e s e q u e n c e s
i n

w i t h


a n d
,
h e n c e
.
B y ( c ), t h e f u n c t i o n

i s c o n t i n u o u s o n
.
T h u s,


.

T h e r e ma i n d e r o f t h e a s s e r t i o n f o l l o w s f
ro
m the same ar
gument.



(e) For
, de
fi
ne
. Prove that (c) and (d) hold for


in place of
. State and
prove an analogue of (b) for

and

in

place of

and
.



S i n c e
,
. B y ( c
),

is

continuous. Let
, t
hen

is
clearly continuous, and so the function

is
also contin
u
ous
.




B y a p
p l y i n g ( d ),
,

, a n d
.




We c l a i m t h a t
i
f

with
,
then

.

Since

implies
,
by
applying (b), we obtain
.