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Question

# Let f(x)=tan(π[x−π])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 x∈Rf(x)=tan(π[x−π])1+[x]2 By definition, [x−π] is an integer, and so (π[x−π]) is an integral multiple of π. ⇒tan(π[x−π])=0 ∀ x∈R Also, 1+[x]2≠0 ⇒f(x)=0 Thus, f(x) is constant function and so, it is continuous and differentiable at all x∈R

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