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Question

If f:RR is a function defined by f(x)=[x1]cos(2x12)π, where [.] denotes the greatest integer function, then f is:

A
discontinuous only at x=1
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B
discontinuous at all integral values of x except at x=1
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C
continuous only at x=1
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D
continuous for every real x
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Solution

The correct option is D continuous for every real x
Doubtful points are x=n, nI
L.H.L.=limxn[x1]cos(2x12)π=(n2)cos(2n12)π=0R.H.L.=limxn+[x1]cos(2x12)π=(n1)cos(2n12)π=0
Also,
f(n)=0
Hence, f is continuous for all real x.

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