Q87 / 200 · WBJEE 2014 · Mathematics

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

  • Correct answer
    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 $x=-2$

Solution

Given, $f(x)=\frac{\tan \left\{\pi\left[x-\frac{\pi}{2}\right]\right\}}{2+[x]^{2}}$ Since. $\left[x-\frac{\pi}{2}\right]$ is an integer for all $x$, therefore $\pi\left[x-\frac{\pi}{2}\right]$ is an integral multiple of $\pi$ for all $x$. Hence, $\tan \left\{\pi\left[x-\frac{\pi}{2}\right]\right\}=0$ for all $x$ Also, $2+[x]^{2} \neq 0$ for all $x$ Hence, $f(x)=0$ for all $x$ Hence, $f(x)$ is continuous and derivable for all $x$.

Hence, the correct answer is option: 'A'.

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