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

$f\left(x\right)\tan x$ is discontinuous

$\text{ when }x=(2n+1)\pi /2,n\in Z$

$f\left(x\right)=x\left[x\right]$ is discontinuous when $x=k,k\in Z$

$f\left(x\right)=\sin \left[n\pi x\right]$ is discontinouswhen $n\pi x=k,k\in Z$

Thus,all the above function have infinite number of points of discontinuity.

But $f\left(x\right)=\frac{\left|x\right|}{x}$ is discontinuous where $x=0$ only.