If f:R→R is a function defined by f(x)=[x−1]cos(2x−12)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,n∈I L.H.L.=limx→n−[x−1]cos(2x−12)π=(n−2)cos(2n−12)π=0R.H.L.=limx→n+[x−1]cos(2x−12)π=(n−1)cos(2n−12)π=0
Also, f(n)=0
Hence, f is continuous for all real x.