Question

For a real number x, let [x] denote the greatest integer less than or equal to x. Then $f\left(x\right)=\frac{tan\left(\pi \right[x-\pi \left]\right)}{1+[x{]}^2}$is:

• A. discontinuous at some x
• B. Continuous at all x but f’(x) does not exist for some x
• C. f’(x) exists for all x but f’’(x) does not exist
• D. f’(x) exists for all x

Answer 1

The correct option is D f’(x) exists for all x

$f\left(x\right)=\frac{tan\left(\pi \right[x-\pi \left]\right)}{1+[x{]}^2}=0\forall x$
$\Rightarrow f\left(x\right),f^{\prime }\left(x\right),f^{\prime \prime }\left(x\right)\Rightarrow =0\forall xϵR$.