0
$\begingroup$

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?

Please help me.Thanks.

$\endgroup$
1
$\begingroup$

A discontinuity of a function is any point where the function is not continuous. This may or may not be a point where the function is defined. You've given examples of both. The person defining the function has the right to choose its domain, and in case of any doubt should state it explicitly.

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.

$\endgroup$
0
$\begingroup$

You have a certain degree of freedom in deciding your domain and codomain, in fact for the latter, it can be hard to establish a consensus. To illustrate this, we can naturally and easily extend the domain and codomain to allow for those points of discontinuity to be included.

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.

The point of this discussion is to demonstrate that your exact question is ambiguous, depending on the context. The domain is usually given to you, if not explicitly, then perhaps an implicit one you should assume throughout the course or textbook. Typically in high-school or AP Calculus level classes, you should not consider infinity (the extra point I added in the above example), except possibly when considering limits. So those points (e.g. vertical asymptotes, which encompasses both 3 and 4) should always be excluded. If you included more context, such as in which course you are considering these questions, or a textbook, I could give a more definitive answer.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.