Differential Calculus with Applications

Six

l if oe I z x if I ex a2

S is not piecewisemonotone

s Sam of 2 monotone truth 9 d h

six will be R2 as a sum of RIfunctions

take gcx3 x non increasing on Eo23 d RI

RiemannIntegrable RI fake huh g

l 0 Excl nondecreasingfunetun

2 et El R2

And gg 1ha g

11 o Ex c six

x 12 I exe2

i six is R2

2 fbeqq d b LCSb P o bonded a b x b

et P Ii Ie In be apartitionofCab

fCx so far x o

Mk sup f o ma int f o for K z n

In Ecc

1st Intervalpatron is 1 a CoXia When of Xi b

and M al m so

since f bat cnetfCxl o For a a a

It followsthat UCfo P Lcf p 0

Thus Ll867 0 and Cfb inf xn aexab o

3 p of Ta c wth U Sb P LL

given forany E 0 thereexist a partitionP of Ea S

lettheahu s L UcfIP L Cfp Cq be fex I show U f L f

Assure UCH FL Cf because Lcf E Cf U p

Thus UCf CC f o Then

Pam'm potea.gs a Ifme

UCJiPJ LCtcPKE Then Ulf EU Cfp c L CfPee LcfHq

Cfl Cc Cf t q L f t Ucf Eff Cf svcs

scuctlabsund.LT Su f is Remain Integrable

sure the function is non deeneisny Has RI

and sinee f es integrableon Ea b Tor e o there exists parthensPi andP2 suchKet

L Cf p Lcf 92 and UcfP2 LUCH442 for P P UP

Left 42 C Lct Pi ELef P220CAP EUCfPzkUcf 142

sure LCH VCH it follows feet UcfPlcLCfpt

hey So idk o exc

fo l 2 x da e x c 2

antidenratee

her x 2 o ex c

2x 42 2 l Exc 2

cheek cont of his

LHC Un has bin x 12 2 42 x 21 x

RHC a bun heat lrn 2 4 2 312

ask T

Yz I 42 s not dictemtrathe

6 g'Cx fix

gCx f Cx fca o c f lb

since fex is increasing f x o

g Lx o

fax oa m magµ

since fca o C f b Tco 0

g cc

local inn at the

4 A funfair is Riemann integrable on a is if

sup L CP fl inf V P f where supremumand

cnf are taken over finitepartitionsof a b

let 2 e a b Suppose heEats R with

h.cz 70 and hot o for all 2CCa b

with X 77

For any P O LCP f U P f so

o is a loner bound for othe setof values taken by LetSz beta lengthof interval that contains z Then UCPf htt Sz Thos v CA f can be madesmaller thanany posture nauhr by choosing a partitionwith 8zsufficiently

small so int UCDf o

This h is R2 and Sa hWdx o

case 2 h Z Lo

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