Question

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

Answer: (A)

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 $x$.