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 value of $x$
• D. discontinuous at $x=-2$

Answer 1

Answer: (A)

continuous for all values of $x$

$[x-\frac{\pi }{2}]$ always gives an integer

$\tan \left(n\pi \right)=0\left(nϵI\right)$

$\Rightarrow \tan \left\{\pi [x-\pi /2]\right\}=0$ always

$\Rightarrow f\left(x\right)=\frac{0}{2+{\left[x\right]}^2}=0(2+{\left[x\right]}^2\ge 2)$

$f\left(x\right)=0=$constant function

$\Rightarrow$ continuous everywhere