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For a real number y, let [y] denotes the...

For a real number y, let [y] denotes the greatest integer less than or equal to y. Then, the function `f(x) = (tan pi[(x -pi)])/(1+[2]^(2)) ` is

A

discontinuous at some x

B

continuous at all x, but the derivative f'(x) does not exist for some x

C

f'(x) exists for all x, but the derivative f'' (x) does not exist for some x

D

f'(x) exists for all x

Text Solution

Verified by Experts

The correct Answer is:
D
Here, `f(x)=(tanpi[(x-pi)])/(1+[x]^(2))`
Since, we know `pi[(x-pi)]=npiandtannpi=0`
`because" "1+[x]^(2)!=0`
`:." "f(x)=0,AAx`
Thus f(x) is a constant function.
`:.f'(x),f''(x)`, . . . .all exist for every x, their value being 0.
`rArr" "f'(x)`exists for all x.
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