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Question

Let [x] denote the greatest integer less than or equal to x. If f(x)=[xsinπx] , then f(x) is :

A
continuous at x=0
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B
continuous in (1,0)
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C
differentiable at x=1
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D
differentiable in (1,1)
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Solution

The correct options are
B continuous at x=0
C continuous in (1,0)
D differentiable in (1,1)
We have 1<x<10xsinπx12
Also xsinπx becomes negative and numerically less
than 1 when x is slightly greater than 1 and so by definition of [x]
f(x)=[xsinπx]=1 when 1<x<1+h
Thus f(x) is continuous and equal to 0 in the closed interval [1,1]
and so f(x) is continuous in the open interval (1,1)
At x=1, f(x) is discontinuous
since limh0(1h)=0 and limh0(1+h)=1
f(x) is not differentiable at x=1

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