Question

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

• A. $f\left(x\right)$ is continuous and differentiable at all $x\in R$
• B. $f\left(x\right)$ is continuous at all $x\in R$ but not differentiable at infinitely many points
  
• C. $f\left(x\right)$ is neither continuous nor differentiable at a finite number of points
  
• D. $f\left(x\right)$ is neither continuous nor differentiable at infinitely many points

Answer 1

The correct option is A $f\left(x\right)$ is continuous and differentiable at all $x\in R$

$f\left(x\right)=\frac{\tan \left(\pi \right[x-\pi \left]\right)}{1+[x{]}^2}$
By definition, $[x-\pi ]$ is an integer, and so $\left(\pi \right[x-\pi \left]\right)$ is an integral multiple of $\pi$.
$\Rightarrow \tan \left(\pi \right[x-\pi \left]\right)=0\forall x\in R$
Also, $1+[x{]}^2\ne 0$
$\Rightarrow f\left(x\right)=0$

Thus, $f\left(x\right)$ is constant function and so, it is continuous and differentiable at all $x\in R$