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Are there any natural examples of removable discontinuities? Most examples I've seen start with a continuous function, and then change the value of a random point, which isn't very natural.

Note that I am not counting examples where the function doesn't exist at the discontinuity ($\frac{\sin x}x$ at 0, for example). I want looking for a natural example of a function $f$ such that for some $c$: $$\lim_{x \rightarrow c} f(x) = L$$ $$f(c)=V$$ $$V \neq L$$

EDIT: This is naturally a , and so somewhat subjective, but I would consider a function "natural" if it isn't defined specifically as an example of a function with a removable discontinuity. The removable discontinuity should be a byproduct, not the goal, of the functions definition.

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  • $\begingroup$ You might want to more adequately define "natural example of a function." The examples you have already seen are probably already what I would call natural examples. Most could be rephrased in terms of using dirac delta functions, indicator functions, heavyside functions, or similar in order to artificially induce the removable discontinuity if what bothers you so much about the normal examples is that they are piecewise defined. $\endgroup$ – JMoravitz Sep 20 '17 at 5:12
  • $\begingroup$ @JMoravitz I attempted to address this in an edit. I'm not sure how precisely I can define it, so maybe this isn't a good question on this site. If it gets closed, and ask on reddit or something. $\endgroup$ – PyRulez Sep 20 '17 at 5:15
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    $\begingroup$ An example of a function which might satisfy you, the function $\text{IsInteger}(x)=\begin{cases}1&\text{if}~x\in\Bbb Z\\0&\text{otherwise}\end{cases}$ can be written instead as $\text{IsInteger}(x)=\lfloor x\rfloor+\lfloor -x\rfloor + 1$. This has removable discontinuities at every integer. $\endgroup$ – JMoravitz Sep 20 '17 at 5:34
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    $\begingroup$ Another fun example, $\text{sgn}(x)=\begin{cases}1&\text{if}~x>0\\0&\text{if}~x=0\\-1&\text{if}~x<0\end{cases}=\lfloor\frac{x}{|x|+1}\rfloor-\lfloor\frac{-x}{|-x|+1}\rfloor$. Using this we can then define $\text{IsZero}(x)=\begin{cases}1&\text{if}~x=0\\0&\text{otherwise}\end{cases}$ as $1-|\text{sgn}(x)|$. We can then further define $\text{Is}_c(x)=\begin{cases}c&\text{if}~x=c\\0&\text{otherwise}\end{cases}$ as $\text{IsZero}(x-c)$. We can then add multiples of $\text{Is}_c(x)$ to any continuous function you want to induce the specific discontinuities. $\endgroup$ – JMoravitz Sep 20 '17 at 5:49
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    $\begingroup$ Thomae's function has a removable discontinuity at every rational point, but I don't know how "natural" it is. $\endgroup$ – bof Sep 20 '17 at 6:46
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I don't know if this is natural for you or not, but here it goes:$$\begin{array}{rccc}f\colon&\mathbb R&\longrightarrow&\mathbb R\\&x&\mapsto&\displaystyle\lim_{n\to\infty}\frac1{1+nx^2}.\end{array}$$It is clear that $f(0)=1$ and also that $x\neq0\implies f(x)=0$.

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