Question

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

• A. continuous everywhere but not differentiable
• B. differentiable once but $f^{\prime \prime }\left(x\right),f^{\prime \prime \prime }\left(x\right),...$do not exist
• C. differentiable for all$x$
• D. discontinuous at integral points

Answer 1

The correct option is C differentiable for all$x$

${\left[x\right]}^2+1\ne 0.$
Also $[x-\pi ]$is always an integer.
$\therefore \pi [x-\pi ]$ is of the form $k\pi$
Hence $f\left(x\right)$ is identically zero for all $x$. So, it is a constant function which is continuous and differentiable for all$x$.