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Question

Consider f(x)=x2;xϵQ and f(x)=1;xQ ; and g(x)=[x]; where [ ] denotes greatest integer function. Then

A
f(x) is continuous at exactly two points
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B
g(x)discontinuous at all xϵI
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C
fog(x) is discontinuous at all xϵQ
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D
gof(x) is continuous everywhere except at x=integer
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Solution

The correct options are
A f(x) is continuous at exactly two points
B g(x)discontinuous at all xϵI
for f(x) to be continuousx2=1x21=0x=1,1
so f(x) is continuous at 2 points
g(x)=[x]
g(x) is discontinuous at all integers because
limxk g(x)=k1
limxk+ g(x)=k
Hence answer is A & B

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