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CONTINUITY AND
DIFFERENTIRBILITY
(ordinuuity
Aunchion flx)^
is
Aacd
o be
sonkinusu
at a , whvu a & domain o f(1) >if
lim_fla)-^
m
,f)
=fla)
a
at
j.e
LH=
RHL
=
Valu o,
a
unthon
at
n=a
im f(z) -fla)
a
lo ntnuBUA hundtion
eal valued
hunttion
JA aid^ ts^ be
dontinuouk , uJt à torhinusus At wey
Pant in the domgain ef
Reson utontinu'ty
4 fl2) is net Lonthinusua atz=a , we ay
dhat fl) Ja discontiruous at 1 a.
*PonibilLha d| tontinuits
lim (^) fla)
and lim^
fla)
Aust but
thuy
au
a
a"
n8t ual,
- im fln) and lt'm fla) au qual but net eQual
-at
te fla).
fla)
i
riot
Asfinud.
ome Basic tontinuRus_untlon
i) Evey
tonsfant
funthdn
ii) every
itentity
untion
ii
uey
Mah'onal unthbn,^
af domain
)Euoy palynomial Junthion
Modulus
funthion f(r) -l«l.
vi)
ALl
Jrigo
nomihri unthon
in thuin
domas'n
ecapt
in And
DAin.
Algubra continugus funchbn.
heorum 1
ut
f
and
be
ho8 ual Ttttmbee.funch'onA
Conh'nuUA
at
A
Hal
numben
t,
thin,
) Cfta)^
Ja
tonhineuBUA
at =c.
i)
Cf-g)
ia
tonhi'nuous
at =C,
ni)
fq
JA
continuBuA
ak
N=C.
)
L
koninusuu
at C^ Mevided
hat
g (r)#0.
Troum m 2
Compositon sf and ia tontinLBUA
et09 nd gof a tonhnupus
e
iuuntiahon. proces o indirg doivative el 1
Dilhpntiability odvability
eal valued hunchon f ik saud to he
dorivable H^ dilhentiahle
at n^ =c
in Jt
domaun, i^
ita
loft
hand and^ ught
hand
dou vaiives at n* C uxt And A LAual Lgpual
t a.
Right hand dvuvah'ue,
f'la) lim fla+h)-fla)
h
h
Aalt hand drivat've.
f'la)
lim
fla-h)
-fla)
h
-h
ThuA,
flx)
ia
ilountiabu
at
2a ,
'}
LRF"la) Lj 'Ca).
Dthuuwise,f(a) a net dillvanhable at -a
RLaabra s Dvwivattvea.
i u#v) =du + dr
d
Aum and dilhuncu
d a
Hule
iCu-v)
ud (^) lv )
vdu) product uls
d
ii)
d(-
d
d (v)^
(quahent Jute)
dx L
whu'u' (^) and '^
au
funthiona .
iv Chain duls
)y4lgla).
dyf'(glx))q'(a).
d
Derivahon
Derivah've
8utsid funthbn.
vnside hunchion.
) flu)
and
o g
(n).
dy. (^) du?
du
d
Dori (^) vahves ef ome (^) atandard unetio
ii)d (r) =nx"i
iAonatant) = 0
i) d (Ainx)* (os
d
v) d^ (wA
N)= -Ain z
dn
vd(ern)=
tnzuhea
is
d dontant
vi ldan n) =sec
da
vi) dAtx)
=
p02 x^ tan .
d
vii (^) ol LOseL N)^
:
dn
i) d (eetx) =rtou 'x dte) =c*
d d
Xi)
(a')
=
a^
Jo9 (^) e a,^ a
0,
xii) (logx) xD
d
rii) (^) d (Jog,) (^) =
,a>o, (^) aFl.
d
Jog a
A n)andond Z^ g(x) be Jwo guven undiona,
De hvtly dihpanhiade beth uncdion h
tesput
Te
n tpealely
nd
tun
put
Jhese
Valuen we get
d dy/d or d d3/dx
d3 d ld
ven
han
d (^) dy ld
Uontàhon o Paramuit unction
flt)
ond =^
gl&)
wher Fis^
a
Panamatr thun dy - dy/d
d
daldt
Doi'vatves dmpliik function
a41l,y)=o
then, d =
i
u.r.t y^
lonsfont)
d
df
Ld
di
f
w.r.f nlwnst
ont)
Loganithmi'c DiHeunhiation
v(x)
funchon
in
or m
u()
in such
CaM
we Jaki loGarithmi'c
and
wAL pHoph'en
oogrihmbimbly di.
Jmpobant
agovikhm fa^
mula
doglmn)- dog^
mt^
log
n
) Jog)= logm - logn
Jog
m
n
dog
m
on bath Jdn
econd 9d Puivafiye.
dy
yfx)
A
qiven
unthon
,
hen
d
f'(x)
is
talu
he
ibat
dvuivahives
of
tallod the astond
flu)and d
den dvu'vahve^
Oyw.r.t, x
andt
ja
dunetd by 4 o
d
Roll a^
te thuoum
4
a
lunehon y
fl)
is
dalineod
n La,
and
unchon (^) f
ia tonhinguaun
jn
la,
b].
ii)
funthon f^
i
dipunhabl
jn la,b)
ii fta) f(b)
Then,
tharu
1nist attuant^
ons
valu
cE (^) la,b)
Auch
that f'()
angnang
Man
Jalu
thuou m.
a
untion
y-
f(x)
ia
dli'ntd
on
La,b]
and
unuhon fis^
condneupun Jn
la,
b7.
unthon f^
a dilunhoblu
n (a,b)
ihin,
Hhou
Likh tluost one^
Jalluu t E( a,b)
Ach hat"(c)-_f{b) -fla)
-a
