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Question

Let f(x)={x[1x]+x[x]if x00if x=0 where [] denotes the greatest integer function, then which of the following is true?


A

f(x) has a removable discontinuity at x=1

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B

f(x) has a non- removable discontinuity at x=2.

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C

f(x) is discontinuous at all positive integers.

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D

None of these

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Solution

The correct options are
A

f(x) has a removable discontinuity at x=1


B

f(x) has a non- removable discontinuity at x=2.


C

f(x) is discontinuous at all positive integers.


f(x)={x[1x]+x[x]if x00if x=0
RHLf(1+)=lima1+(x[1x]+x[x])=limx1(0)+x(1)
LHL=limx1(x[1x]+x[x])=limx1(1)+x(0)=1
f(1)=1[11]+11=2.
Removable Discounting at x=1


RHL=limx2+x[1x]+x[x]=2(0)+2.2=4
LHL=limx2[1x]+x[x]=2(0)+2.1=2
LHL RHL. Limit does not exist.
Non removable at x=2


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