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Question

The functionf(x)=[x]cos[(2x-1)/2]π,[.] denotes the greatest integer function is discontinuous at


A

All x

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B

All integer points

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C

No x

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D

x which is not an integer

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Solution

The correct option is A

All x


Explanation for the correct option

f(x)=[x]cos(2x-1)2π

For, a function to be continuous

L.H.L=R.H.L=f(x)

Let the 0x<1

f(x)=[x]cos(2x-1)2π,0x<1

Solving L.H.L

limx0-f(x)limx0-[x]cos2x-12π0×cos-12π0

Solving R.H.L

limx1+f(x)limx1+[x]cos2x-12π[1]cos2-12πcosπ2=0

Now, f(x)=f(0)

f(x)=[x]cos(2x-1)2π

f(0)=0cos-12π

f(0)=0

L.H.L=R.H.L=f(x)

So, f(x) is continuous for all x.

Hence, the correct option is A.


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