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So, I have been classifying in my mind Hole discontinuities as removable, piecewise, or "non-cancelling" (for example the limit at zero of $\frac{\sin{x}}{x}$)

I have done this by associating the term removable with the act of cancelling out, therefore removing a discontinuity to obtain an equivalent function that is continuous at that same point, and whose graph is identical everywhere to the original function except at the hole. For "non-cancelling" type, we could still argue that the function written as a power series would cancel the denominator, and result in an equivalent continuous function.

However, in the case of a piecewise function that is simply defined to be missing a point, there is no mathematical operation that will remove said discontinuity.

But I recently found in the openstax calculus book, and the college board description of discontinuities, that hole discontinuities are simply treated as removable for all cases. Which seems weird to me to have 2 names for exactly the same mathematical object, instead of removable being just a subcategory of hole discontinuities. Is there any definition I am missing?

I guess my question is, is there a formal, standardized definition for hole/removable discontinuities? Since so many books seem to treat the "removability" as a matter of cancelling out factors, and that is obviously not what's happening in the examples I gave above.

Update: The wiki article on classification of discontinuities says "The term removable discontinuity is sometimes broadened to include a removable singularity, in which the limits in both directions exist and are equal, while the function is undefined at the point $x_0$ .This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's domain."

Now, wikipedia not being the best of sources, is there a more reliable formal source that explains and possibly agrees with this abuse of terminology assessment?

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    $\begingroup$ The discontinuity of $\frac{\sin(x)}{x}$ at $x = 0$ is removable. $\endgroup$
    – Ricky
    Aug 31, 2023 at 7:09
  • $\begingroup$ "But I recently found that hole discontinuities are simply treated as removable for all cases"...where? $\endgroup$
    – 5xum
    Aug 31, 2023 at 7:29
  • $\begingroup$ There are many notions in math based on the ability to express something in a certain way, such as a rational number being a real number that can be expressed as the quotient of two integers. However, what you're talking about is not one of these -- there seems to be nothing mathematically useful in classifying holes in terms of the possibility of algebraic manipulations of this sort. Depending on the author, a hole is simply a point where the limit exists and the value of the function is different OR a point where the limit exists and the value of the function is different or doesn't exist. $\endgroup$ Aug 31, 2023 at 8:45
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    $\begingroup$ A point at which a function is not continuous is a removable discontinuity exactly when you can change the value at only that point and get a function that is continuous at that point. There is nothing more to it. $\endgroup$ Sep 1, 2023 at 5:17
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    $\begingroup$ In particular, whether you can cancel something or not is entirely irrelevant. $\endgroup$ Sep 1, 2023 at 5:19

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The function $$f(x)=\frac{\sin x}{x},\quad x\ne 0$$ is continuous. In a sense, it is incorrect (or abuse of terminology) to say that it is discontinuous at $x=0$, since $x=0$ is outside of its domain. It is better to say that it is undefined at $x=0$.

Now the function $$g(x)=\begin{cases}\frac{\sin x}{x} &\text{for } x\ne 0 \\ 2 &\text{for } x=0\end{cases}$$ is discontinuous at $x=0$. This discontinuity is removable.

Different people use words differently. I would say the graph of $f$ has a hole (but again, technically, it is not a discontinuity).

For both $f$ and $g$, it is natural to proceed instead to the function $$h(x)=\begin{cases}\frac{\sin x}{x} &\text{for } x\ne 0 \\ 1 &\text{for } x=0\end{cases}$$ which is continuous everywhere.

Starting from $f$, we arrive at $h$ by enlarging/extending the domain, without ruining the continuity. Starting from $g$, we arrive at $h$ by removing a discontinuity, but we do so by changing the value at a point.

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