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Question

Let f(x)=tan(π[xπ])1+[x]2, where [.] denotes the greatest integer function. Then

A
f(x) is continuous and differentiable at all xR
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B
f(x) is continuous at all xR but not differentiable at infinitely many points
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C
f(x) is neither continuous nor differentiable at a finite number of points
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D
f(x) is neither continuous nor differentiable at infinitely many points
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Solution

The correct option is A f(x) is continuous and differentiable at all xR
f(x)=tan(π[xπ])1+[x]2
By definition, [xπ] is an integer, and so (π[xπ]) is an integral multiple of π.
tan(π[xπ])=0 xR
Also, 1+[x]20
f(x)=0

Thus, f(x) is constant function and so, it is continuous and differentiable at all xR

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