Natural examples of removable discontinuities?
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 soft-question, 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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$\begingroup$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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