Question

Which of the following functions have finite number of points of discontinuity in R ([.] denotes the greatest integer function)?

• A. $\tan x$
• B. $x\left[x\right]$
• C. $\frac{|x|}{x}$
• D. $\sin \left[n\pi x\right]$

Answer 1

Answer: (C)

$\frac{|x|}{x}$

$tan\left(x\right)$ has infinite number of discontinuities i.e. at every $\frac{n\pi }{2}$ where$n$ is an integer

$x\left[x\right]$ is discontinuous at all integers
$\frac{|x|}{x}$ is discontinuous where$|x|$ is discontinuous i.e. $0$

$sin\left[n\pi x\right]$ is also discontinuous at infinite points.

$\therefore \frac{|x|}{x}$ is only discontinous at $0$.

So the required function is $\frac{|x|}{x}$