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Question

If f(x)=cosπ(|x|+[x]), then f(x) is/are (where [] denotes greatest integer function)

A
continuous at x=12
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B
continuous at x=0
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C
Differentiable in (2,4)
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D
Differentiable in (0,1)
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Solution

The correct options are
A continuous at x=12
D Differentiable in (0,1)
We go about doing this problem considering each option.
(a) f(12)=cosπ2=0
f(12+)=cosπ2=0. Hence option a is correct.
(b) f(0+)=cos(0)=1,f(0)=cos(π)=1. Hence option b is incorrect.
(c) f(x) is discontinuous at x=3 and thus not differentiable in the given interval.
(d) In (0,1),f(x)=cosπx which is continuous and differentiable.

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