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VED1
E L-LOVELY-H math2-1 IInd 21-10-11 IIIrd 24-1-12 IVth 21-4-12 Vth 20-8-12 VIth 10-9-12

noEPk;soh dk rfDs

B'N i/eo fe;h fpzd{ x = a s/ cbB brksko j? sK dk wkB, x B{z a d/ pj[s B/V/ u[D
e/ fJZSk nB[;ko S'Nk ehsk ik ;edk j?.

2H10 fe;h fpzd{ T[Zs/ cbB dh brkskosk
(Continuity of a Function at any Point)
e'Ph gfoGkPk (Cauchy’s definition): fJZE cbB f (x), x = a T[Zs/ brksko ejkT[Adk j?.
i/eo fJZe ;t/S (arbitrary) u[Dh rJh XBkswe ;zfynk ε fijVh fezBh th S'Nh j't/ gozs{
Iho' Bk j't/, d/ bJh n;hA ε T[Zs/ fBoGo fJZe nfijh XBkswe ;zfynk δ nfijh gqkgs
eo ;ehJ/ fe d/ T[jBK ;ko/ wkBK d/ bJh fiBQK d/ bJh

H

Gkt nzsokb (a – δ, a + δ) d/ fe;h th fpzd{ x T[Zs/ cbB f (x) ns/ f (a) dk XBkswe nzso fJZe
;t/Ssk fBdfoPN XBkswe ;zfynk ε s'A th xZN j't/, sK cbB f (x) fpzd{ x = a T[Zs/ brksko
ejkT[Adk j?.
t?ebfge gfoGkPkL cbB f (x), x = a T[Zs/ brksko (Continuous) ejkT[Adk j? i/eo dk
ti{d j't/ ns/ T[j x = a T[Zs/ cbB d/ wkB d/ pokpo j't/.

B'N; f (x) = f (a) Gkt [f (x) dh ;hwk, id'A ] & [cbB dk wkB, id'A x = a]

fJ; soQK i/eo f (a + 0) = f (a – 0) = f (a), sK f (x) B{z x = a T[Zs/ brksko efjzd/ jB. BjhA sK
x = a T[Zs/ cbB nbrksko j?. cbB f (x) fpzd{ x = a T[Zs/ brksko ejkT[Adk j? fe i/eo cbB
fBwBfbys fszB PosK dh g[PNh eo/ L

1H i/eo cbB f (x), x = a T[Zs/ gfoGkfPs j't/ Gkt x = a T[Zs/ cbB dk fJZe fBPfus wkB j't/.
2H id'A x dk wkB a d/ tZb nZr/ tZXdk j't/, T[d'A f (x) fe;h ;hwk d/ tZb nZr/ tX/ Gkt f
(x) ti{d j't/.
3H f (x) = f (a) j't/.

2H11 brkskosk dk i:kfwsh noE (Geometrical Meaning of

Continuity)

wzB b' QR cbB f (x) dk b/ykfuZso j?. fJ; b/ykfuZso
ftZu fpzd{ P[a, f (a)] T[Zs/ ftuko eo'. i/eo ε e'Jh
;t/Ssk fBdfoPN XBkswe ;zfynk j? sK o/yktK y = f
(a) – ε, y = f (a) + ε, x-neP d/ ;wkB ns/ P d/
fBZub/ gk;/ ns/ T[Zgo d/ tZb j't/rh.

fdZs/ j'J/ ε d/ ;zrs x = a d/ gfos n;hA fJZe nzsokb
fi;dh u"VkJh 2δ j?, gsk eo ;ed/ jK fi;B{z o/yk x
= a d/ d'B'A gk;/ ns/ y-neP d/ ;wkB o/yktK x = a –
δ ns/ x = a + δ d[nkok gqdofPs ehsk ik ;edk j?.

48 LOVELY PROFESSIONAL UNIVERSITY