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Question

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

A
discontinuous at integral points
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B
continuous everywhere but not differentiable
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C
differentiable once but f′′(x),f′′′(x),...do not exist
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D
differentiable for all x
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Solution

The correct option is D differentiable for all x
[x]2+10.
Also [xπ]is always an integer.
π [xπ] is of the form kπ
Hence f(x) is identically zero for all x. So, it is a constant function which is continuous and differentiable for all x.

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