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Question

If f(x)=[tanx]+tanx[tanx],0x<π2, where [.] denotes thegreatest integer function, then

A
f(x) is continuous in [0,π2)
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B
f(x) is not continuous at x=0
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C
f(x) is continuous at x=0,π4
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D
f(x) has infinite points of discontinuity
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Solution

The correct options are
A f(x) is continuous in [0,π2)
B f(x) is continuous at x=0,π4
f(x)=[tanx]+tanx[tanx]=[t]+t[t],,
where t=tanx.
Clearly, 0t< at 0x<π2.
Possible points of discontinuity may be, at which tN.
Let t=kN
L.H.L. at t=k=limtk[t]+t[t]
=limh0[kh]+(kh)[kh]
=limh0{k1+[khk+1]}=k
R.H.L. at t=k=limtk+[t]+t[t]
=limh0[k+h]+k+h[k+h]
=limh0k+k+hk=k
The function is continuous at t=kN
Thus, function f(x) is continuous for all [0,π2)

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