Which of the following functions have finite number of points of discontinuity?

Finite number of points of discontinuity.

From Option-A: $\tan x$ is continuous on R.

Option-B: $x\left[x\right]$ is discontinuous at integers.

Option-C: $\frac{\left|x\right|}{x}$ is discontinuous at only $x=0$.

Option-D: $\cot x$  is discontinuous at $x=\{n\pi /n\in Z\}$.