23
NAMe .viane3h Re.NO : 3ssaoP leol6 CLA SS I. M. Se CmaT4) Sub hnAralysi - DATE25 (tl2oa Ica Si rdt t Sindt cot)Sdt Sindt 2 S3-c omma ha d -1 Tha 9tas Tm et On R.H. betom 20 S7tt) ndt

THIRUPATHY M

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Real Analysis II model exam

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Page 1: THIRUPATHY M

NAMe .viane3h Re.NO : 3ssaoP leol6 CLA SS I. M. Se CmaT4)

Sub hnAralysi -

DATE25(tl2oa

Ica

Si rdt t Sindt

cot)Sdt

Sindt 2

S3-c omma ha d -1

Tha 9tas Tm et On R.H.

betom 20

S7tt) ndt

Page 2: THIRUPATHY M

c4

tntegal ep&ntaiion for Tha DesinRn

Sum of a Souvio evie Pto 4 iad

) oeh

C C)

TO Pooe afundion ¬L C) Suth ra3

(.du) C Sa

n S

Sn-Sm

C

mi

m

C4 u) i ni

C Cu Ci)

Sn Sl Gelce

auth Setun CT)

Page 3: THIRUPATHY M

Cu

YC(d».du)

C n)

Cs.de)- cf.d) n-) tAhe-4

Cn).de

1341 -C.Qu

Cte de) Ce

(eho

Rim S-S -o

142 1141 Cu0 od n

Page 4: THIRUPATHY M

Gniven ta ron ond Onnees

Poin

pol ona nin,

ord P

Sa mon Qor in

,2. .

a.Tt D;+1 -itPe TtePA P

2 of jo;-)-CP:)o

e uations

fce - fto ) fte)-ttP, )* +f R)-F(P

3 wuteant Oh 3

eorem,

2ntl

Page 5: THIRUPATHY M

l2 (2, 2

Delin aralvalusd fonion folm

h 2) dot CD: (2i)) 14 )

Sunticn oniroruj

2n Da 44ig

SbirO TRO h Tto) toE hinu

wed hon houJ

On B(o)

A SSUm ot Covtar ontaditt ton)

fca 4ft) for Som poipoinr

Poy hemn-yd ot

n 3

ThRoron

y;(2') C3-z) 9)-4o co for P\,a. .

y-fo)-o (9y-CiCaJ»o :i t.e.

Co 1 2

hosU DT (2i)fo

ForALh?

Page 6: THIRUPATHY M

On1Odio hion C i ch

Ps- on

4 n

PostT AS Samu That

mcacoriclR

Aivnon eo pen Sat , d G2

S.Ccen 2e'caz

E tb 1G

ehoe Snt able 7o St

om. 6 Adding n both St

e ul 4t2)

C +l2 ce+ eE

wwett te r n

Page 7: THIRUPATHY M

A fain Ect E'c2

EUE' e Grura

,0l

la. T

Prol - That o7 iren 0+ tu p0

G, & Ga StEco, RE'ct2 d COrn2) ceG

TO Ps0 othrra by

Eca and e ca2

E S( and e G2

S ItA2l+6

Sin ES co sat gon

moasusabO

Hen

Page 8: THIRUPATHY M

pominatad Conv«7 en ThsDm Leb c

tatument:.

Sen un of fanins be

L s. S.t d in n on cx almost evet

where Cns x46) SuPPox

n s zo) Caxsb) nET hen

S.+ fEC.b1

tim n

feL Ta.Lj

Cti en be e a Seq can u ot That

Untion n Ca.L3

TO POve

Seuent o meal urask

untion on to bI a

Page 9: THIRUPATHY M

Im

a nouon ThRD em

Than s mealurabu daY

- 4o0 3 Ao ien

EL Las

fe Cab]

Hen R P7ov ci)

b-

CoTollat ( well noðn

E, E o Lab

measMT able Snbs

ECE2 c

meas uTb

Than EN?'s N

Page 10: THIRUPATHY M

Po

n

Cven That EL D.b

Lmm wo noun

GtvRn

141 1

int e N.9t

me-Cb-al c

-a) menled

b- -me

Page 11: THIRUPATHY M

Fn-Ss S

Cn -)|6

SnS

Hanl prond

Page 12: THIRUPATHY M

S % Cn o3m+bn sinna)

fereier

b

ic f-selstf-62

fc-Snta | dx

Hoym end'nunu lo97

e o.aFJ

2

2T -

tc tha SnfL? (6 2t T. do>n-1 2

On o 23

Sn

)

Page 13: THIRUPATHY M

2 CaSn Cx)1*dx0

Cev fouTiQ paced3 oda

co-cid ct ffe2 Do 2n gire

T6 PoN

te hc

hox fud

fox xe CoT ad n -1. 2

post n14dollov R

SinCR

n h

hn ht) dnifom on Con

ftDdt n

outoJk1 bu inttdt

colt S

Co Le C3ut b Stn tdt

n iloy ml io

Page 14: THIRUPATHY M

To

Suppoe fo Caunio Soyi

not n - bnSim)

Do.a3 TRon sn ConYJe)

Lee ho th n( convjod Fog

4han6A9y no Co herss Ao Soma

Sam imv Co B« O fo)

folod tkal

SnCx

n CohyeTZotor o m

ConveTSeJ

Page 15: THIRUPATHY M

slat and ghean Gulo

Stament

ASSmn Tal

+ota eTivotine 8'c Lo b-go Qnd

aSSum& Tha d: ffovontiabt o* to oth

4cia cirinalivo f'cb en Shovo that The

Compos it Sunction Sa d:ftsenttab lo. f fotal

na c)ata) Lincar unton A )

CoPOS ion o

8'ca)

To PTOve funRion h-fq

ad cotaX driia

ca cL a ca) Proe That

h COHg - hta)+T (8 Ea hne

ane

Eay

NOO ha-8 ha)

Page 16: THIRUPATHY M

Otiren

oon iao ofb

S

ayo-Sosmudon

GtOn C6 to

hco) -ha)=}C)(u )4el ecv) hooR Fb)-s0

hco -o)-f CbocY)-+CYil Csco)

her o

Gten tdad

odes toj to :

ScuetsAios

Page 17: THIRUPATHY M

h Cos)ha f°c6) La' caay) (y E aty

4l vlEb (u)

whone S0 h

Hon ThA proot

Slab and Pe mie fondi0n

Statmet

Co A e Subse of r an ha : Ase ha ceninuous a+HaX

n

ASsum

n A I Cw#0.zinA Tha Sho tal Oponn OPen

Given

A an bpen Sub ot p an

Sbe Snaot of Ain eh

Page 18: THIRUPATHY M

s

to ho coriuo parAta A53rm

darivaiv darvaiva 4 o on and oian 4

AgA.cnalu ing hon

t on Ca')

Con LO B)

Apn s

Ca) 3 ao opO CA opern

toaeo.c Aen

Page 19: THIRUPATHY M

Stats od paov TO nt dor mula Auntdion

Statment AS Sum That an

of od Am ane QSAOTent?otble Arinaliyo

pot f pOtn+ of S.

on Open Sof Pn d

Such at on lio a aach

an CAne b uo

om Ca.b) Such To

m)

(ba)+

ivcn an pen Subet tn p'

eoIem f. a fo Screh TFot

atl a fn intrva engen

o

enuai 3) taa t cb-a] 3en C-, ) he opoite

aco fce)

fCo)

dine 2ioral Ta lo nla ko

Cc)Cn-< + o2ce)+

Page 20: THIRUPATHY M

.Ca

lC0, n-m Jubin

mH

L

)

wrtlAn a n

ThR chain ula to

CPLL t)

(PAs) (E -ot)

- (t) CPCAD; b-a)

CC:M) (e)u = Af Cut)

A2on chafnTula The

continui T PTOR we hovMR

Cm)

* F Oom hoome

fCb f)=a c -gco

h= L m

(Pb)a) ¢( ptt):b-a)

C- roa)

Page 21: THIRUPATHY M

Ca

Cnincn Tho

fCa.bS andC SubsetB

And moasuT obe

A Ont Cvle-G.O

pei 7uia oi me aun eqble

En-En)u

mCeuév) 2

CEn- En-1) u

2

disjoint mo aduTenbla

Fn maasuseqb Covolod na

TeEu-S -mE-me 5 2

2

LE -¬u-S En -mË nA

e-2

FTomO

m fn) mE Aim Lmen-man-

n-S

mEn n

Proot . HRnR

Page 22: THIRUPATHY M

Griven TRad S boondod noaobo

An n on Ca.. 4mo

la bl-O mC m Yx La.bT

iven Eofa finite namber of poin

S o m Jo 9.9.

Sui 3icn 01 C m)

Sush Tha R

Suceas Poin e 3bdiy Son is

lan

fo each 2

t

Sin mRadurabl? unuic

E mcaurol

on Eu e E En mounob

Potibion m S:ET b ) E] J.u. fy

Page 23: THIRUPATHY M

xeEu

umeu from

CSPT meu (fom

Cr3

ME

Ca)

B wa nown e eLtT

theorO