Question

Let $f\left(x\right)=\frac{\sin 4\pi \left[x\right]}{1+[x{]}^2}$, where [x] is the greatest integer less than or equal to x, then

• A. $f\left(x\right)$ is not differentiable at some points
• B. $f\left(x\right)$ exists but is different from zero
• C. $LHD(atx=0)=0.RHD(atx=1)=0$
• D. $f\left(x\right)=0$ but $f$ is not a constant function

Answer 1

The correct option is C $LHD(atx=0)=0.RHD(atx=1)=0$

$f\left(x\right)=\frac{sin4\pi \left[x\right]}{1+[x{]}^2}$
$\left[x\right]\to integer$
$\therefore sin4\pi \left[x\right]=0$
$\therefore f\left(x\right)=0$
$\therefore f^1\left(x\right)=0$