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Question

Let [x] be the greatest integer less than or equal to x. Then at which of the following point(s) the function f(x)=xcos(π(x+[x])) is discontinuous

A
x=1
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B
x=0
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C
x=1
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D
x=2
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Solution

The correct option is D x=2
f(x)=xcos(π(x+[x]))
At x=0,
L.H.L.=f(0)=0R.H.L.=f(0+)=0
So, f(x) is continuous at x=0
At x=1,
L.H.L.=f(1)=1cos(π(1+0))=1R.H.L.=f(1+)=1cos(π(1+1))=1
So, f(x) is discontinuous at x=1
At x=1,
L.H.L.=f(1)=1
R.H.L.=f(1+)=1
So, f(x) is discontinuous at x=1
At x=2,
L.H.L.=f(2)=2
R.H.L.=f(2+)=2
So, f(x) is discontinuous at x=2

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