Question

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

• A. Continuous for all values of $x$
• B. Discontinuous at $x=\frac{\pi }{2}$
• C. Not differentiable for some values of $x$
• D. Discontinuous at $x=-2$

Answer 1

The correct option is A Continuous for all values of $x$

Given,

$f\left(x\right)=\frac{\tan \left\{\pi [x-\frac{\pi }{2}]\right\}}{2+{\left[x\right]}^2}$

Since, $[x-\frac{\pi }{2}]$ is an integer for all $x$,

$\therefore \pi [x-\frac{\pi }{2}]$ is an integral multiple of $\pi$ for all $x$.

Hence, $\tan \left\{\pi [x-\frac{\pi }{2}]\right\}=0$ for all $x$

Also, $2+{\left[x\right]}^2\ne 0$ for all $x$.

Hence, $f\left(x\right)=0$ for all $x$.

Hence, $f\left(x\right)$ is continuous and derivable for all $x$.