Question

The function $f\left(x\right)=\frac{\tan \left(\pi \right[x-\pi \left]\right)}{1+{\left[x\right]}^2}$, where [x] is the greatest integer function,

• A. Is discontinuous at some x
• B. $f\prime \left(x\right)$ exists for all x
• C. $f\prime \left(x\right)$ exists for all x but $f{\prime }^{\prime }\left(x\right)$ does not exist
• D. Is continuous for all x but $f\prime \left(x\right)$ does not exist for some x

Answer 1

The correct option is B $f\prime \left(x\right)$ exists for all x

$f\left(x\right)=\frac{tan\left(\pi \right[x-\pi \left]\right)}{1+[x{]}^2}$
$tan\left(\pi \right[x-\pi \left]\right)$ where $[x-\pi ]=x$
than $tan\left(n\pi \right)=0$
$\therefore f\left(x\right)=0$
$\therefore f^1\left(x\right)=0$