Problems in Mathematical Alnalysis III Kaczkornowak

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Problems in
Mathematical
Analysis HI
Integration

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STUDENT MATHEMATICAL LIBRARY
Volume 21

Problems in
Mathematical
Analysis III
Integration
W. J. Kaczor
M.T. Nowak

#AMS

AMERICAN MATHEMATICA L SOCIET Y

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Editorial Boar d
David Bressoud , Chai r Danie

l L . GorofT Car

l Pomeranc e

2000 Mathematics Subject Classification. Primar y 00A07 , 26A42 ;
Secondary 26A45 , 26A46 , 26D1 5 , 28A1 2 .
For additiona l informatio n an d updates o n this book , visi t
www.ams.org/bookpages/stml-21
Library o f Congres s Cataloging-in-Publicatio n D a t a
Kaczor, W . J . (Wieslaw a J.) , 1 949 [Zadania z analizy matematycznej . English ]
Problems i n mathematica l analysis . I . Rea l numbers , sequence s an d serie s /
W. J . Kaczor , M . T. Nowak .
p. cm . — (Studen t mathematica l library , ISS N 1 520-91 2 1 ; v. 4)
Includes bibliographica l references .
ISBN 0-821 8-2050- 8 (softcove r ; alk. paper )
1. Mathematica l analysis . I . Nowak , M . T . (Mari a T.) , 1 951 - II . Title .
III. Series .
QA300K32513 200 0
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Printed i n the United State s o f America .
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Contents

Preface vi

i

Part 1 . Problem s
Chapter 1 . Th e Riemann-Stieltje s Integra l 3
§1.1. Propertie s o f th e Riemann-Stieltje s Integra l 3
§1.2. Function s1
o f Bounde d Variatio n

0

§1.3. Furthe r Propertie s o f th e Riemann-Stieltje s Integra
1l

5

§1.4. Prope r Integral s 2 1
§1.5. Imprope r Integral s 2

8

§1.6. Integra l Inequalitie s 4

2

§1.7. Jorda n Measur e 5

2

Chapter 2 . Th e Lebesgu e Integra l 5

9

§2.1. Lebesgu e Measur e o n th e Rea l Lin e 5

9

§2.2. Lebesgu e Measurabl e Function s 6

6

§2.3. Lebesgu e Integratio n 7 1
§2.4. Absolut e Continuity , Differentiatio n an d Integratio n 7

9

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Contents

VI

§2.5. Fourie r Serie s 8

4

Part 2 . Solution s
Chapter 1 . Th e Riemann-Stieltje s Integra l 9

7

§1.1. Propertie s o f th e Riemann-Stieltje s Integra l 9

7

§1.2. Function s 1
o1
f Bounde d Variatio n

4

§1.3. Furthe r Propertie s o f the Riemann-Stieltje s Integra1l 2

6

§1.4. Prope r Integral s 4

3

1
§1.5. Imprope r Integral s 6

4

§1.6. Integra l Inequalitie s 20

7

§1.7. Jorda n Measur e 22

8

Chapter 2 . Th e Lebesgu e Integra l 24

7

§2.1. Lebesgu e Measur e o n th e Rea l Lin e 24

7

§2.2. Lebesgu e Measurabl e Function s 26

8

§2.3. Lebesgu e Integratio n 28 1
§2.4. Absolut e Continuity , Differentiatio n an d Integratio n 29

6

§2.5. Fourie r Serie s 3

6

Bibliography - Book s 35 1
Index 35

5

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Preface

This i s a seque l t o ou r book s Problems in Mathematical Analysis I,
II (Volume s 4 an d 1 2 i n th e Studen t Mathematica l Librar y series) .
The boo k deal s with th e Riemann-Stieltje s integra l an d th e Lebesgu e
integral fo r rea l function s o f one rea l variable . Th e boo k i s organize d
in a wa y simila r t o tha t o f the firs t tw o volumes , tha t is , i t i s divide d
into tw o parts : problem s an d thei r solutions . Eac h sectio n start s
with a numbe r o f problem s tha t ar e moderat e i n difficulty , bu t som e
of th e problem s ar e actuall y theorems . Thu s i t i s no t a typica l prob lem book , bu t rathe r a supplemen t t o undergraduat e an d graduat e
textbooks i n mathematica l analysis . W e hop e tha t thi s boo k wil l b e
of interes t t o undergraduat e students , graduat e students , instructor s
and researche s i n mathematical analysi s an d it s applications. W e also
hope tha t i t wil l b e suitabl e fo r independen t study .
The first chapte r of the book is devoted to Riemann an d Riemann Stieltjes integrals . I n Sectio n 1 . 1 w e conside r th e Riemann-Stieltje s
integral wit h respec t t o monotoni c functions , an d i n Sectio n 1 . 3 w e
turn t o integratio n wit h respec t t o function s o f bounde d variation .
In Sectio n 1 . 6 w e collec t famou s an d no t s o famou s integra l inequal ities. Amon g others , on e ca n fin d OpiaP s inequalit y an d Steffensen' s
inequality. W e clos e th e chapte r wit h th e sectio n entitle d "Jorda n
measure". Th e Jorda n measure , als o calle d conten t b y som e authors ,

vn

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Vlll

Preface

is no t a measur e i n th e usua l sens e becaus e i t i s no t coun t ably addi tive. However , i t i s closely connecte d wit h th e Rieman n integral , an d
we hope that thi s section will give the student a deeper understandin g
of the idea s underlyin g th e calculus .
Chapter 2 deals with the Lebesgu e measur e an d integration . Sec tion 2. 3 present s man y problem s connecte d wit h convergenc e theo rems tha t permi t th e interchang e o f limi t an d integral ; LP spaces o n
finite interval s ar e als o considere d here . I n th e nex t section , absolut e
continuity an d the relation between differentiation an d integration ar e
discussed. W e present a proo f o f th e theore m o f Banac h an d Zareck i
which state s tha t a function / i s absolutely continuou s o n a finite in terval [a , b] i f and onl y i f it i s continuous an d o f bounded variatio n o n
[a, b], an d map s set s o f measur e zer o int o set s o f measur e zero . Fur ther, th e concep t o f approximate continuit y i s introduced. I t i s worth
noting her e that ther e i s a certain analog y betwee n tw o relationships :
the relationshi p betwee n Rieman n integrabilit y an d continuity , o n
the on e hand , an d th e relationshi p betwee n approximat e continuit y
and Lebesgu e integrability , o n th e othe r hand . Namely , a bounde d
function o n [a , b] i s Rieman n integrabl e i f an d onl y i f i t i s almost ev erywhere continuous ; an d similarly , a bounde d functio n o n [a , b] is
measurable, an d s o Lebesgu e integrable , i f an d onl y i f i t i s almos t
everywhere approximatel y continuous . Th e las t sectio n i s devoted t o
the Fourie r series . Give n th e existenc e o f extensiv e literatur e o n th e
subject, e.g. , th e book s b y A . Zygmun d "Trigonometri c Series" , b y
N. K . Bar i " A Treatis e o n Trigonometri c Series" , an d b y R . E . Ed wards "Fourie r Series" , w e foun d i t difficul t t o decid e wha t materia l
to includ e i n a boo k whic h i s primaril y addresse d t o undergraduat e
students. Consequently , w e hav e mainl y concentrate d o n Fourie r co efficients o f function s fro m variou s classe s an d o n basi c theorem s fo r
convergence o f Fourie r series .
All the notatio n an d definition s use d i n this volume ar e standard .
One ca n find the m i n the textbook s [27 ] and [28] , which als o provid e
the reade r wit h th e sufficien t theoretica l background . However , t o
avoid ambiguit y an d t o mak e the boo k self-containe d w e start almos t

every sectio n wit h a n introductor y paragrap h containin g basi c defi nitions an d theorem s use d i n th e section . Ou r referenc e convention s

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Preface

IX

are bes t explaine d b y th e followin g examples : 1 .2.1 3 or I , 1 .2.1 3 o r
II, 1 .2.1 3 , whic h denot e th e numbe r o f th e proble m i n thi s volume ,
in Volum e I o r i n Volum e II , respectively . W e als o us e notatio n an d
terminology give n i n th e first tw o volumes .
Many problem s hav e bee n borrowe d freel y fro m proble m section s
of journals lik e the America n Mathematica l Monthl y an d Mathemat ics Today (Russian) , an d fro m variou s textbooks an d proble m books ;
of thos e onl y book s ar e liste d i n th e bibliography . W e woul d lik e t o
add tha t man y problem s i n Sectio n 1 . 5 com e fro m th e boo k o f Ficht enholz [1 0 ] an d Sectio n 1 . 7 i s influence d b y th e boo k o f Rogosinsk i
[26]. Regrettably , i t wa s beyon d ou r scop e t o trac e al l th e origina l
sources, an d w e offer ou r sincer e apologies if we have overlooked som e
contributions.
Finally, w e woul d lik e t o than k severa l peopl e fro m th e Depart ment o f Mathematics o f Maria Curie-Sklodowsk a Universit y t o who m
we are indebted . Specia l mentio n shoul d b e mad e o f Tadeusz Kuczu mow an d Witol d Rzymowsk i fo r suggestion s o f severa l problem s an d
solutions, an d o f Stanisla w Pru s fo r hi s counselin g an d Te X support .
Words o f gratitud e g o t o Richar d J . Libera , Universit y o f Delaware ,
for hi s generou s hel p wit h Englis h an d th e presentatio n o f th e ma terial. W e ar e ver y gratefu l t o Jadwig a Zygmun t fro m th e Catholi c
University o f Lublin , wh o ha s draw n al l th e figure s an d helpe d u s
with incorporatin g the m int o th e text . W e than k ou r student s wh o
helped u s i n th e lon g an d tediou s proces s o f proofreading . Specia l
thanks g o t o Pawe l Sobolewsk i an d Przemysla w Widelski , wh o hav e

read th e manuscrip t wit h muc h car e and thought , an d provide d man y
useful suggestions . Withou t thei r assistanc e som e errors , no t onl y ty pographical, coul d have passed unnoticed . However , we do accept ful l
responsibility fo r an y mistake s o r blunder s tha t remain . W e woul d
like t o tak e thi s opportunit y t o than k th e staf f a t th e AM S fo r thei r
long-lasting cooperation , patienc e an d encouragement .
W. J . Kaczor , M . T . Nowa k

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Part 1

Problems

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Chapter 1

The Riemann-Stieltje s
Integral

1.1. Propertie s o f th e Riemann-Stieltje s Integra l
We star t wit h som e basi c notations , definition s an d theorems . B y a
partition P o f a close d interva l [a , b] we mea n a finite se t o f point s
xo, xi,..., xn suc h tha t
a = XQ < x\ < .. . < x n-\ <
The numbe r n(P) = maxjx ^ — xi-\ :
raes/i of P.

x n = b.

i = 1 , 2 , . .. ,n} i s calle d th e

For a functio n a monotonicall y increasin g o n [a , b] w e writ e
A ^ = a(xi) - a(xi-i).
If / i s a real function bounde d o n [a , 6], we define th e uppe r an d lowe r
Darboux sum s o f / wit h respec t t o a an d relativ e t o P , respectively ,
by
nn

U(P,f,a) =

Y,MiAa t, L(P,f,a)

=

^m^Aa, ,

where
Mi = su

p /(#)

, ^

i = in

f /(#)

•
3

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4

Problems. 1 : Th e Riemann-Stieltje s Integra l

We als o pu t
pb rb

/ fda = mfU(P if,a), /

fda = supL(P , f,a),

J aJ

a

where th e infimu m an d th e supremu m ar e taken ove r al l partition s P
of [a, 6], and call them, respectively, the upper an d the lower Riemann Stieltjes integral . I f th e uppe r an d th e lowe r Riemann-Stieltje s inte grals are equal, we denote the common valu e by / fda an d cal l it th e
Riemann-Stieltjes integra l o f / wit h respec t t o a ove r [a , b]. I n thi s
case w e sa y tha t / i s integrabl e wit h respec t t o a , i n th e Rieman n
sense, an d w e writ e / G 1Z(a). I n th e specia l cas e o f a(x) = x w e
get th e Rieman n integral . I n thi s cas e th e uppe r (lower ) Darbou x
sum correspondin g t o a partitio n P , an d th e uppe r (lower ) Rieman n
integral ar e denoted , respectively , b y U(P,f) ( L ( P , / ) ) , an d f
(fafdx) .

afdx

The Rieman n integra l o f / ove r [a , b] i s denoted b y j a fdx.

Moreover, correspondin g t o ever y partitio n P o f [a , b] w e choos e
points t i , t 2 , . . . , t n suc h tha t Xi-\ < t{ < Xi, i = 1 ,2 , . . . , n , an d
consider th e su m
n

5(P,/,a) = 53/(ti)Aai.
2 =1

We say tha t

lim S(PJ,a)

=

A

if, fo r ever y e > 0 , ther e i s S > 0 suc h tha t /i(P ) < S implie s tha t
|5(P, /, a) — A\ < e fo r al l admissibl e choices o f U. I n th e cas e whe n
a(x) — x w e se t

5(P,/) = £/(*i)(x i -x < _ 1 ).
i=l

Throughout thi s section , / i s always assume d t o b e bounde d an d
a monotonicall y increasin g o n [a , b]. I n th e solution s w e will often us e
the followin g theorem s (see , e.g. , Rudi n [28]) .
Theorem 1 . / G 71(a) on [a , b] if and only if for every e > 0 there
exists a partition P such that

U(PJ,a)-L(PJ,a)
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1.1. Propertie s o f th e Riemann-Stieltje s Integra l 5
Theorem 2 . If f is continuous on [a,b], then f G 71 (a) on [a , b\.
1.1.1. Suppos e a increase s o n [a , 6], a < c < b, a i s continuou s a t
c, f(c) — 1, an d f(x) — 0 fo r x ^ c . Sho w tha t / G lZ(a) an d tha t
/ * / * * = <>.

1.1.2. Suppos e / i s continuou s o n [a , 6], a < c < fr, a(x ) = 0 i f
x G [a, c), an d a(x) = 1 if x G [c, 6]. Sho w tha t J* a / da = /(c) .
1.1.3. Le t 0 < a < b and
fx

ifar€[a,6]nQ,

\o ifxG[a,6]\Q

.

Find th e uppe r an d lowe r Rieman n integral s o f / ove r [a , 6].
1.1.4. Le t a > 0 an d
(x ifx G [ - o , o ] n Q ,
[0 i f x G [ - a , a ] \ Q .
Find th e uppe r an d lowe r Rieman n integral s o f / ove r [—a , a].
1.1.5. Sho w tha t th e so-calle d Rieman n functio n
0i
f{x) = {l/q i

f x i s irrational o r x = 0 ,
f x = p/q, p G Z , g r G N , an d
p, g are co-prime ,

is Riemann integrabl e o n ever y interva l [a , b].
1.1.6. Le t / : [0,1] - + R b e define d b y settin g

[0 otherwise .
Show tha t J Q f(x)dx = 0.
1.1.7. Sho w tha t / : [0,1] - > R define d b y

\x ~ ix] otherwis

e

is Riemann integrabl e o n [0,1 ] .

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6

Problems. 1 : Th e Riemann-Stieltje s Integra l

1.1.8. Defin e
JO if * G [-1,0], /
/(x) = < an
d a(x
\ l i f x G ( 0 , l ] , \l
Show tha t / G 7£(a ) althoug h li m S(P,

O if * G [-1,0),
)= <
iixe
[0,
1 ] .
/ , a) doe s no t exist .

1.1.9. Sho w tha t i f / an d a hav e a commo n poin t o f discontinuity i n

fa, 61 , then li m S(P, f.a) doe s no t exist .
LJ

'M

(P)-o v

)

1.1.10. Prov e tha t i f li m S(P,

/ , a) exists , the n / G 1 1 (a) o n [a , 6]

/x(P)-0

and
lim 5(P ; , /, a) = / / d a .

M(P)-O V

J

a

Show als o tha t fo r ever y / continuou s o n [a , 6], th e abov e equalit y
holds.
1.1.11. Sho w tha t i f / i s bounded an d a i s continuous o n [a , 6], the n
/ G 1 Z(a) i f an d onl y i f li m S(P, / , a) exists .
MP)-o
1.1.12. Le t

(c i f a < x < x*,
a(x) = <
[d i f x * < x < 6 ,
where c < d an d c < a(x*) < d. Sho w tha t i f / i s bounde d o n [a , 6]
and suc h tha t a t leas t on e o f th e function s / o r a i s continuous fro m
the lef t a t x * an d th e othe r i s continuou s fro m th e righ t a t x*, the n
/ G 1 1 (a) an d
C

f(x)da(x) =

f(x*)(d-c).

Ja

1.1.13. Suppos e that / i s continuous o n [a , 6] and a i s a step function
that i s constant o n th e subinterval s (a , ci), (ci, C2),..., (c m , 6), wher e
a < c\ < C2 < • - • < c m < 6. Sho w tha t
b ™>

/
f(x)da(x) =

/(o)(a(a+ ) - a(a)) + £ /(c

fc )(a(c+)

- a(c fe"))

fc=i

+ /(6)(a(6)-a(fe")) .

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1.1. Propertie s o f the Riemann-Stieltjes Integra l 7
1.1.14. Usin g Rieman n integral s o f suitabl y chose n functions , find
the followin g limits :
(a) li m ( +
+
•• • + — ),
v;
n-o o \n + 1 n + 2 3nJ
ini n2
3 + -" +
( b ) Jn-^oo
„y n ( ^ r3 T+-l^3 +n ^3 -+ T2 ^
n

(c) h m ^-

j,

k

3

- f n3 / '

> 0,

(d) li m - ij/( n + l)( n + 2)---( n + n),
n—>oo 77 ,
/ 77

, 77

, 77

,

(e) li m si n +i2 i 2 n sin H2 + 2 2 n h
+
ol/n
sin • 2 + n 2
n r

<2

/

(f) lin ^m
( + —
— + ••• + + l / n
ooyn+1 n
+ 1/2 n
y

(g) li m y/f(l/n)f(2/n)... / ( n / n )

, wher e / i s a positiv e an d

n—»oo

continuous functio n o n [0,1 ] .
1.1.15. Sho w tha t th e limit
/ si n - ^r si n -%- si
n -^ \
Um n ±
l + 5±
L + . . . + »±
i
n-^oo 1
12
U
I
is positive .
1.1.16. Sho w tha t i f / i s continuously differentiabl e o n [0,1 ] , the n

/(D-/(0)
Using thi s result , calculat e
/ fc
fc
n /c
/ l + 2 + -- - +1
hm n —
,
f c +i

n->oo y

n

A

\
C+ 1

fc > 0.
/

1.1.17. Fo r k > 0, calculat e
fc
fc
l i m / l + 3 + -- - + ( 2 n - l )

n—>oo \ 7 7 /

fc

^

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Problems. 1 : Th e Riemann-Stieltje s Integra l

8

1.1.18. Suppos e tha t / i s twic e differentiabl e o n [0,1 ] an d / " i s
bounded an d Rieman n integrable . Sho w tha t

^y(£mdx-±±f

2i-i\\ _ r ( i ) - r ( o
^ 2n 2

)

4

1.1.19. Fo r n G N , defin e

1
1

J

_

and
2

2

2n + 1 2

n+3 4

n- 1

Show tha t
lim [7 n = li m V n — I n 2.
n—>oo n—

->oo

Moreover, usin g th e result s state d i n 1 .1 .1 6 and 1 .1 .1 8 , sho w tha t
lim n(In 2 — U n) — - an
n-^oo 4

d li

n—>o

m n 2 (ln2 — V n) — —.
o 3

2

1.1.20. Sho w tha t i f / i s Rieman n integrabl e ove r [a , 6], the n / ca n
be change d a t a finit e numbe r o f point s withou t affectin g eithe r th e
integrability o f / o r th e valu e o f it s integral .
1.1.21. Sho w tha t i f / i s monotoni c an d a i s continuou s o n [a , 6],
then / G 71(a).
1.1.22. Prov e tha t i f / G 71 (a) an d a i s neither continuou s fro m th e
left no r fro m th e righ t a t a point i n [a , 6], then / i s continuous a t thi s

point.
1.1.23. Le t / b e Rieman n integrabl e an d a continuou s o n [a , b]. I f
a i s differentiabl e o n [a , b] except fo r finitel y man y point s an d a' i s
Riemann integrable , the n / G IZ(a) an d
pb pb

/ f(x)da(x) =

Jo..

/ f(x)a'(x)dx.
Ja

1.1.24. Le t / b e Rieman n integrabl e an d a b e continuou s o n [a , b]
except fo r finitel y man y points . I f a i s differentiabl e o n [a , b] except

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1.1. P r o p e r t i e s o f t h e Riemann-Stieltje s Integra l 9
for finitely man y point s an d a' i s Riemann integrable , the n / G 7Z(a)
and
pb pb

/ f(x)da{x)= f{x)a'{x)dx
J a

+

f{a){a{a

+

)-a{a))

J a
m

+ £ f(c

k)(a(4)

-

a(c~)) + f(b)(a(b) - a(b~)),

k=l

where c^ , k = 1 , 2 , . . ., m, ar e point s o f discontinuit y o f a i n (a , b).
1.1.25. Calculat e j_ 2x2da{x), wher

e

x+ 2 i

f -2
a{x) = < 2 i

f-

2

x + 3 i f0

,

1 < x < 0,
< x < 2.

1.1.26. Prov e th e Fzrs t Mean Value Theorem. I f / i s continuous an d
a i s monotonicall y increasin g o n [a , 6], the n ther e i s c G [a , 6] suc h
that
rb

I

f(x)da(x) =

f(c)(a(b)-a(a)).

J a

1.1.27. Sho w tha t i f / i s continuou s an d a i s strictl y increasin g o n
[a, 6], the n i t i s possible t o choos e c G (a, 6) suc h tha t th e equalit y i n
the firs t mea n valu e theore m (state d above ) holds .
1.1.28. (a ) Le t / b e continuou s o n [0,1 ] . Fo r positiv e a an d b, find
the limi t
lim /

/

l^ldx.

J as

(b) Calculat e

f1 x n
lim / dx.
n
^°° Jo 1 + x
1.1.29. Suppos e / i s continuous an d a i s strictly increasin g o n [a , &];
define
F(x)= f

f(t)da(t).
J a

Show tha t fo r x G [a, b]
+ ft) - F ( s )
v F(x
lim — ; ^ 7^f = f(x) .
h-o a( x + ft) - a(x )

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Problems. 1 : Th e Riemann-Stieltje s Integra l

10

1.1.30. Suppos e / i s continuou s o n [a , 6], a i s bot h continuou s an d
strictly increasin g there , an d th e limi t
+ h)-f(x) df
r f(x
lim — —
— = —{x)
h-^o a(x + h) — a(x) da
exists an d i s continuou s o n [a , b]. Show tha t
rb

df
-£(x)da(x) =

f(b)-f(a).

1.2. Function s o f Bounde d Variatio n
Recall tha t th e total variation V(f; a , b) of / o n [a , b] i s

V(f;aS=snplf2\f(xi)-f(xi.1)\\,
where th e supremu m i s take n ove r al l partition s P = {xo,xi , ...,x n }
of [a , 6]. I f V{f\ a , 6) < +oo , the n / i s said t o b e of bounded variation
on [a , b]. We als o defin e
Vf(x) = V(f; a , x), a

< x < b.

Clearly, Vf(a) = 0 an d v y i s monotonicall y increasin g o n [a,b\. Th e
following theore m say s tha t a functio n o f bounde d variatio n ca n b e
exhibited a s a differenc e o f tw o monotoni c functions .
Theorem 1 . / / / is of bounded variation on [a , b], then
f(x) - f{a) =p{x) -q(x),
where
P(x) = \(v f(x) +

f(x)-f(a)) and

q(x)

= \{v f{x) -

j\x) + /(a) )

are monotonically increasing on [a , b].
The function s p an d q are calle d th e positive an d negative variation functions of f, respectively .
1.2.1. Sho w tha t th e functio n give n b y
fx2cos^ if*G(0,l]
J V T/

' I

iss

0i

,

fx - 0

differentiabl e o n [0,1 ] bu t no t o f bounde d variation .

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1.2. F u n c t i o n s o f B o u n d e d V a r i a t i o n

11

1.2.2. Sho w t h a t i f / ha s a bounde d derivativ e o n [a , 6], then / i s o f
bounded variation .
1.2.3. Sho w t h a t th e functio n
_, A f *
/ ( X )=

2

cosf i f z e ( 0 , l ]

\ 0 if

,

*= 0

is o f bounde d variatio n o n [0,1 ] .
1.2.4. Sho w t h a t

v(/;M) =/(&)-/(<*)
if an d onl y i f / i s monotonicall y increasin g o n [a , b].
1.2.5. Fo r a e R an d / ? > 0 , defin e
iixe (0,
1 ]
it x = 0 .

,

Show t h a t / i s o f bounde d variatio n i f an d onl y i f a > (3.
1.2.6. Sho w t h a t i f / i s o f bounde d variatio n o n [a , b], the n / i s
bounded o n [a , b).
1.2.7. I f / an d g ar e o f bounde d variatio n o n [a , 6], the n s o i s thei r
product fg. Moreover , i f in f \f(x)\ > 0 , the n g/f i s als o o f bounde d
x(z[a,b]

variation o n [a , b].
1.2.8. Mus t th e compositio n o f tw o function s o f bounde d variatio n
be o f bounde d variation ?
1.2.9. I f / satisfie s a Lipschit z conditio n an d g i s o f bounde d varia tion, the n th e composit e functio n / o g i s o f bounde d variation .
1.2.10. Sho w t h a t i f / i s o f bounde d variatio n o n [a , 6], the n s o i s
|/|P, l < p < + o o .
1 . 2 . 1 1 . Prov e t h a t i f / i s continuou s o n [a , b] and | / | i s o f bounde d
variation o n [a , 6], the n s o i s / . Prov e als o t h a t continuit y i s a n
essential hypothesis .
1.2.12. I f / an d g ar e o f bounde d variatio n o n [a , 6], then s o i s h(x) =

max{/0),#0)}.

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Problems. 1 : Th e Riemann-Stieltje s Integra l

12

1.2.13. W e sa y tha t / : A— > R , A C M , satisfies a Holder condition
(also calle d th e Lipschitz condition of order a) o n A i f ther e exis t
positive constant s M an d a suc h tha t
\f(x)-f(x')\
r x,x'eA.

(a) Sho w tha t th e functio n

f(x) = f*dm
V

' \

if

-e(0,
1 /2],

o if

x= 0

is of bounde d variatio n o n [0,1 /2 ] an d doe s no t satisf y a Holde r con dition.
oo

(b) Set x n = Y2 k in 2 fc' n — 2 , 3 , . .. . Let / b e the continuous functio n
k=n

on [0 , X2] define d a s follows :
/(0) = /(*„ ) = 0 , f(^±^±^j=

1

-, n

= 2,3,... ,

and / i s linear o n [rr n+i, (x n + x n+i)/2] an d [(x n-\-xn+i)/2,xn}. Prov e
that / satisfie s a Holde r conditio n fo r ever y 0 < a < 1 , an d tha t / i s
not o f bounde d variatio n o n [O,^] 1.2.14. Suppos e that / : [a, 00)— > M i s of bounded variatio n o n every
interval [a , 6], b > a , an d pu t
V ( / ; a , o o ) = li m V(f;a,b).
b—>oo

Show tha t i f V(f; a , 00) < 00 , the n th e finite limi t li m f(x) exists .
x—*oo

Does th e opposit e implicatio n hold ?
1.2.15. Fo r / define d o n [a , b] an d a partitio n P — {xo,x'i,.. . , x n }
of [a , 6], w e for m th e su m

vup)=x;i/(si)-/(zi-i)iProve tha t i f / i s continuou s o n [a , 6], the n
lim V ( / , P ) = V(/:a,6) ,
that is , fo r an y s > 0 there exist s 6 > 0 such tha t //(P ) < < 5 implie s
V(f;a.b)~V(f,P)
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1.2. F u n c t i o n s o f B o u n d e d V a r i a t i o n

13

1.2.16. Fo r / define d o n [a , b] and a partitio n P = { x o , x i , • • • ,x n}
of [a , 6], we for m th e su m
n

W(f,P) = Y,(Mr-m t),
2=1

where
M% = su

p /(x)

, ra* =

in

f f(x).

X&[Xi-.i,Xi] X(zlXi-i,Xi]

Using th e resul t i n th e previou s problem , sho w t h a t i f / i s continuou s
on [a , 6], the n
lim W(f,P)
M(P)-0

W

=
'^

V(f;a,b).
^

'}

1.2.17. Suppos e t h a t / i s o f bounde d variatio n o n [a , b], with p an d q
its correspondin g positiv e an d negativ e variatio n function s a s define d
in Theore m 1 . Suppos e als o t h a t p\ an d q\ ar e increasin g function s
on [a , b] such t h a t f = pi — q\. Sho w t h a t i f a < x < y < b, the n
p(x) - p(y) < pi(x) -

pi{y) an

d q(x)

- q{y) < qi(x) - q^y).

Conclude t h a t V(p; a , b) < V(jp\\ a , b) an d V(q\ a , b) < V{q\\ a , b).
1.2.18. Suppos e t h a t / i s o f bounde d variatio n o n [a , b] and f(x) >
m > 0 , x G [a , 6]. Sho w t h a t ther e ar e tw o monotonicall y increasin g
functions g an d / i suc h t h a t

/?,(x)

1.2.19. Comput e th e positiv e an d negativ e variatio n function s o f
(a) f(x)=^-\x\, a : G [ - l . l ]
(b) f(x)

=

cosa: , x

(c) / ( ! • ) = J - M . . r

,

€ [0.2TT] ,

e [0.3] .

1.2.20. Assum e t h a t / i s o f bounde d variatio n o n [a , b]. Prove t h a t i f
/ i s continuou s fro m th e righ t (left ) a t TQ , the n Vf i s als o continuou s
from th e righ t (left ) a t TQ.
1 . 2 . 2 1 . Sho w t h a t th e se t o f point s o f discontinuit y o f a functio n
/ o f bounde d variatio n o n [a J)] i s a t mos t countable . Moreover , i f

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14

P r o b l e m s . 1 : T h e Riemann-Stieltje s Integra l

{xn} i s the sequenc e o f points o f discontinuity o f / , the n th e functio n
g(x) = f(x) — s(x), wher e s(a) = 0 an d
s(x) = f(a+) - f(a) + £ (/(*+

) - f(x~)) + f(x) - / ( * " )

Xn
for a < x < 6 , i s continuou s o n [a , 6]. (Th e functio n s i s calle d th e
salius function of f.)
1.2.22. Le t / b e o f bounde d variatio n o n [a , b] an d le t
x
1f
f(t)dt,
x— a Ja

g(a) = 0 an d g(x ) = /

E (a, 6].

x

Prove tha t g i s of bounde d variatio n o n [a , 6].

1.2.23. Sho w that i f / satisfie s a Lipschitz conditio n o n [a , 6], then t>/
also satisfies th e Lipschitz conditio n with the same Lipschitz constant .
1.2.24. Prov e tha t i f / i s o f bounde d variatio n o n [a , b] and enjoy s
the intermediat e valu e property , the n / i s continuous. Conclud e tha t
if / ' i s o f bounde d variatio n o n [a , 6], the n f i s continuous .
1.2.25. Prov e tha t i f / i s continuously differentiat e o n [a , 6], the n
X

«/(*)= [

\f'(t)\dt.

Ja

1.2.26. Sho w tha t i f / i s continuous an d a monotonicall y increasin g
on [a , 6], the n th e functio n
F(x) = / f(t)da(t),

[a , b],

xe

./a

is o f bounde d variatio n o n [a , 6].
1.2.27. I f / O ) = li m f n(x) fo r x G [a, b], the n
n—->oo

V(f;a,b)< li

m V(/„;a,b) .
n—• oo
oo o

1.2.28. Suppos e tha t th e serie s ^T

a

o

n a n d Yl b n ar e absolutel y con -

n=l n=

l

vergent, an d le t {x n} b e a sequenc e o f distinc t point s i n (0,1 ) . Prov e
that th e functio n / define d b y

/(0)=0, f(x)=

Yl

a

- + J2

b

^ fo

r

*e(0,l ]

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