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The standard $\varepsilon$-$\delta$ definition of a limit of real-valued functions goes as follows:

Let $f\colon A \to \mathbb{R}$ be a function where $A\subseteq \mathbb{R}$, and let $c$ be a limit point of $A$. Then $\lim_{x\to c}f(x) \to L$ if $\forall \varepsilon > 0, \exists \delta > 0$ such that $\forall x\in A,$ $0<\lvert x-c\rvert <\delta \implies \lvert f(x)-L\rvert < \varepsilon$.

I believe this is the standard limit definition. However, I have seen in several places an alternative definition that - instead of requiring $c$ to be a limit point as above - demands that the domain $A$ of the function contain a punctured neighbourhood of $c$.

My issue is that I'm not sure these definitions are equivalent. If anything, the latter definition appears to be more restrictive that the former; for instance, consider the square root function. It is well-known that $\sqrt x$ is continuous at $x=0$. But if one resorts to using the second definition, $\lim_{x\to 0}\sqrt x$ would actually be undefined because the domain does not contain a punctured neighbourhood around $0$.

So, my question is: is there any (perhaps historical?) motivation for using the second definition? Have I maybe missed something? Thank you.

Chubby Chef
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  • For the punctured neighborhood definition, there is a further requirement that you need to intersect it with $A$. So for $\sqrt{x}$ for example, you can still take $U = (-\delta, \delta)$ around $0$ but you only consider points in $[0, \infty) \cap U$. Topologically speaking, $[0, \infty) \cap U$ is the punctured neighborhood of $0$ in the subset topology of $[0, \infty)$. – balddraz Apr 14 '21 at 15:28
  • The punctured nbd definition was, I believe, earlier, and it is generally used in the context of calculus to make things simpler and prevent issues with weird functions that would create exceptions that would not be illuminating (and would instead be distracting). The more general definition appears in Bourbaki; I believe it did not arise until the notions of topology started being more developed. – Arturo Magidin Apr 14 '21 at 15:42
  • Somewhat related: this – Arturo Magidin Apr 14 '21 at 15:44
  • @ArturoMagidin thank you for the info and the reference - if you wish to expand it into an answer I'll accept it. On a tangential note, would you happen to have a good history of mathematics book recommendation (that does not shy away from actual mathematics) that deals with the development of rigorous analysis (i.e. Cauchy-Weierstrass and later) in a substantial manner? – Chubby Chef Apr 14 '21 at 19:28

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You're correct that the first definition ("limit point") is implied by the second ("punctured neighborhood"), that the sole difference is the nature of the domain of $f$ near $c$, and that they're not equivalent.

I'm not a historian, but believe the second definition originates with differentiation (which limits make precise), where one wants $c$ to be an interior point of $A$, in which case the difference quotient $\frac{f(x) - f(c)}{x - c}$ is defined in a punctured neighborhood of $c$.

Andrew D. Hwang
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  • To elaborate on why differentiability requires an interior point: in higher dimensions, the derivative may not be unique if we allow simple limit points. – Vercassivelaunos Apr 15 '21 at 11:11