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, $2+[x{]}^2\ne 0$ for all $x$
Hence, $f(x)=0$ for all $x$.
Hence, $f(x)$ is continuous and derivable for all $x$.