Given f(x)={3−[cot−1(2x3−3x2)]forx>0{x2}cos(e1x)forx<0 (where {} and [] denotes the fractional part and the integral part functions respectively). Then which of the following statements do/does not hold good? f(0−)=0 b. f(0+)=3 c. if f(0)=0 , then f(x) is continuous at x=0 d. irremovable discontinuity of f at x=0
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f(x)=[x]
2
+
x−[x]
at x=1
x→1
−
lim
f(x)=
x→1
−
lim
(0
2
+
x+0
)
as (x→1
−
;0<x<1;[x]=0)
=
1
=1
x→1
+
lim
f(x)=
x→1
+
lim
(1
2
+
x−1
)
=1+0=1
as x→1
+
⇒1<x<2
[x]=1
LHL = RHL =f(1)
at x=n,nϵI−{1}
x→n
−
lim
f(x)=(n−1)
2
+
n−(n−1)
=(n−1)
2
+1
=n
2
−2n+2
x→n
−
lim
f(x)=n
2
+
n−n
=n
2
(n−1)
2
−1
=n
2
⇒ LHL
= RHL
∴ Discontinuous as xϵI−{1}
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