THEORY OF REAL FUNCTIONS

THEORY OF REAL
FUNCTIONS
HAN D WRITT EN
NOT E S




BY SANJIBON HAZARIKA

, a amafion
Continuity
esn muous or
whathar tha wnzio
diseoriimuows.

-

ufl) - fa)
LHL RHL


h ies beTweon O amd 1

+oC
oh
Sequenea

f:N R
13 equal to eo-domaim
Romge
O unelion -
Diferanctiablit of
On inloeval


I =a,b]e R R also m inorval

A nor fo) 4harwntiaWa at a
pant C on I - [o 6 L f ) - - Rf)
whara,
Lf)- m
e-0


t ) - fl)
amd
Rfe) = +0 -e

, O what esmdunon a wne[ion on



inteva i dharontiabla
Let a e b

Lfe) - Rf )
At
Rf exist

At b
Lfb) exut
2. A wnefion i defined O R

#) =J , itox<

cheek whathor h a w.ation
deuivabla O mo ehoek Hha darivability
ha wmEtionm

So -
Give f)

At
fn) - f4)
9-o


-




= 1

, Jim
+o -1

Jn

= O




:Lf')# RfD
i n mot darivalla t - I

A fwmetion i dafnad
) - f *'sin ) , if a4o
=o

darivaba ot 0, but

fo) 4 f(%) SAarivat.ive Lunetion
no eerinou
at


Sol
G
fx)- an{),if
o

At -0,


Rf -0
sin - o
J