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Question

For a real number y, let [y] denote the greatest integer less than or equal to y. Then the function f(x)=tan π[(xπ)]1+[x]2

A
discontinuous at some x
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B
continuous at all x, but the derivative f'(x)does not exist for some x
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C
f'(x) exists for all x, but the derivative f''(x) does not exist for some x
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D
f'(x) exists for all x
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Solution

The correct option is D f'(x) exists for all x
Here, f(x)=tanπ[(xπ)]1+[x]2
Since, we know π[(xπ)]= nπ and tan nπ=01+[x]20f(x)=0,x
Thus, f(x) is a constant function.
f(x), f"(x), ........ all exist for every for x, their value being 0.
f(x) exists for all x.

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