# The points where function is discontinuous,are those points counted/considered in the domain of the function.

The points where function is discontinuous,are those points counted/considered in the domain of the function.
$(1)[x]$,greatest integer function is discontinuous at all integer points but integers are considered in the domain of $[x]$
$(2)x-[x]$,fractional part function is discontinuous at all integer points but integers are considered in the domain of $x-[x]$
$(3)\tan x$ is discontinuous at $\frac{\pi}{2},\frac{3\pi}{2}$ and these points are not considered in the domain.
$(4)$If a function has a hole/vertical asymptote at a point,then that point is not considered in the domain.
$(5)$The points where function has jump discontinuity,are those points considered part of the domain.

I want to ask in which type of discontinuities ,the points of discontinuity are considered part of domain and in which type of discontinuities ,the points of discontinuity are not considered part of domain?

Of course the function is not defined at a point, then it certainly can't be continuous there, because the definition of continuity at $x=a$ requires that the function is defined at $a$.
In many cases, when we define a function by a formula we implicitly assume the domain is the set of all $x$ for which the formula makes sense. That may or may not include the discontinuities.
Consider Example 4 in the case of a vertical asymptote by looking at perhaps the easiest example: $f(x) = 1/x$. The author from whom you quote would probably consider this as a function $f: \mathbf{R} - 0 \to \mathbf{R}$ (by this notation, I mean its domain is the real numbers except zero and its codomain is all real numbers.) However, we could extend this to all of $\mathbf{R}$ by simply extending the codomain to a point, which we define to be $\{\infty\}$ and just define $f(0) = \infty$. Now, we can consider $f: \mathbf{R} \to \mathbf{R} \cup \{\infty\}$ where I have extended both the domain and codomain by a point.