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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

fJekJh^2 L ;hwk ns/ brkskosk

i/eo d/ o/yktK x = a - δ, x = a + δ d/ ftZu cbB f (x) d/ b/ykfuZso dk jo fpzd{ d' o/yktK y = a - B'N
ε, y = a + ε d/ ftukb/ th j't/ sK cbB f (x) fpzd{ x = a T[Zs/ brksko j?.
T[dkjoD L (i) nuo cbB (constant function) f (x) = c, x d/ jo/e n;b wkB d/ bJh brksko
j?.
(ii) cbB f (x) = sin x ns/ f (x) cos x, x d/ jo/e n;b wkB d/ bJh brksko j?.

2H12 fe;h fpzd{ T[Zs/ cbB dh brkskosk gsk eoB dh ftXh (Method to
Finding Continuity of a Function at any Point)

;hwk (limit) dh gfoGkPk s'A ;gPN j? fe lim f (x) dk ti{d (existence) sK jh j[zdk j?, id'A
x→ a
f (x) dh tkw gZy ;hwk & f (x) dh dZyD gZy ;hwk
Gkt lim f (x) = lim f (x)
x→ a - x→ a +
Gkt f (a – 0) = f (a + 0)
fJ; soQK fpzd{ x = a T[Zs/ cbB f (x) dh brkskosk (continuity) gqdofPs eoB d/ bJh ;kB{z
fdykT[Dk ukjhdk j? fe fJ; fpzd{ T[Zs/ f (x) dh tkw gZy ;hwk = f (x) dh dZyD gZy ;hwk &
cbB dk wkB
Left Hand Limit = Right Hand Limit = Value of the Function
Gkt lim f (x) = lim f (x) = f (a)
x→ a − x→ a +
iK f (a – 0) = f (a + 0) = f (a)
dZyD gZy ;hwk (R.H.L.) d/ bJh f (x) ftZu x = a + h oZy', fiZE/ h → , id'A x →
a
0
tkw gZy ;hwk (L.H.L.) d/ bJh f (x) ftZu x = a - h oZy', fiZE/ h → , id'A x →
a
0

2H13 fJZe nzsokb ftZu cbB dh brkskosk
(Continuity of a Function in an Interval)
fJZe cbB f (x) fe;h fttfos nzsokb (a, b) ftZu brksko fejk iKdk j? i/eo fJj nzsokb
(a, b) ftZu x d/ ;ko/ wkBK d/ bJh brksko j?.
cbB f (x) fe;h pzd nzsokb (closed interval) [a, b] ftZu brksko fejk iKdk j?, i/eo
(i) fJj x d/ T[jBK ;ko/ wkBK d/ bJh brksko j't/ fi;d/ bJh a < x < b

(ii) lim f (x) = f (a)
x→ a+ 0
(iii) lim f (x) = f (b)H
x→ b- 0
Gkt cbB y[Zb/ nzsokb (a, b) ftZu brksko j't/ ns/ x = a T[Zs/ dZyD gZy s'A ns/ x = b T[Zs/ tkw
gZy s'A brksko j't/.

fJZe nzsok ftZu nbrkskosk (Discontinuity in an interval)L cbB f (x) fe;h nzsokb ftZu
nbrksko fejk iKdk j? i/eo fJj nzsokb ftZu fe;h fJZe fpzd{ T[Zs/ iK pj[s ;ko/ fpzd{nK T[Zs/
nbrksko j't/.

2H14 brksko cbBK dk gqw/: (Theorem on Continuous Functions)

(i) i/eo f (x) ns/ g (x) d'B'A fe;h fpzd{ x = a T[Zs/ brksko j'D sK f (x) ± g (x) th x = a
T[Zs/ brksko j't/rk.

LOVELY PROFESSIONAL UNIVERSITY 49