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From Wikipedia:

Suppose $X,Y$ are topological spaces with $Y$ a Hausdorff space. Let $p$ be a limit point of $Ω⊆X$, and $L ∈Y$. For a function $f : Ω → Y$, it is said that the limit of $f$ as $x$ approaches p is L (i.e., $f(x)→L$ as $x→p$) and write $$ \lim_{x \to p}f(x) = L $$ if for every open neighborhood $V$ of $L$, there exists an open neighborhood $U$ of $p$ such that $f(U∩Ω- \{p\}) ⊆ V$.

I wonder if people also often generalize the definition of a limit of a function $f$ to the case when $p$ is an isolated point of $\Omega$?

Can the above definition except that $p$ is a limit point of $\Omega$ can be applied to the case when $p$ is an isolated point of $\Omega$?

Specifically, is the "openness" of $V$ wrt the topology of $X$ or wrt the subspace topology on $\Omega$?

Thanks and regards!

Ali Caglayan
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Tim
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    If $p$ is an isolated point, then we can always find $U$ such that $U\cap\Omega-p = \varnothing$, in which case the condition $f(U\cap\Omega-p)\subseteq V$ is trivially satisfied. This means that $\lim\limits_{x\to p}f(x) = L$ for all $L$, which makes the notion of "limit" as an isolated point rather useless. (Intuitively: the limit asks what the function $f$ is doing "near", but not at the point $p$; if $p$ is isolated, then there is no "near" there). – Arturo Magidin Feb 14 '12 at 21:47
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    As to your last question: doesn't matter: because $A\subseteq \Omega$ is open with respect to the topology on $\Omega$ if and only if there is a set $V$ that is open in $X$ such that $V\cap \Omega = A$. Since wee are only considering what happens inside $\Omega$, whether you look at $A$ or at $V$, you get the same intersection. – Arturo Magidin Feb 14 '12 at 21:50
  • The process of removing the point $p$ smells a bit fishy to me, if only because of this kind of ambiguities. The definition in the French Wikipedia and in my topology courses do not use this: we want $f(\Omega \cap U) \subset V$. Is this a French/English difference, or the English Wikipedia using a weird definition? – D. Thomine Feb 14 '12 at 22:03
  • @D.Thomine: nice question. I also want to know why – Tim Feb 14 '12 at 22:07
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    @D.Thomine: The problem with that definition is that functions that are defined but not continuous at $p$ cannot have a limit at $p$. The entire notion of "removal discontinuity" goes away. – Arturo Magidin Feb 14 '12 at 22:08
  • @D.Thomine: The difference is probably inherited from Bourbaki; a quick perusal of their General Topology reveals that they define limits in terms of filters, and consider cases like $\lim\limits_{x\to p,x\neq p}f(x)$ separately from $\lim\limits_{x\to p,x\in \Omega}f(x)$. – Arturo Magidin Feb 14 '12 at 22:13
  • @Arturo: there is a notion of "limite épointée" (I don't know how to translate this) for this purpose, with a notation like $\lim_{x \to p, x \neq p} f(x)$. [Edit: thank you for the Bourbaki reference; I would have been surprised if the usual French definition were distinct from the Bourbaki one.]. – D. Thomine Feb 14 '12 at 22:20
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    @D.Thomine: Just like the Bourbaki definition I mentioned. But, no, the English Wikipedia is not using a "weird definition", in the sense that it is a standard definition in topology/analysis books. For example, all real analysis books with which I am familiar that define limits in terms of $\epsilon$-$\delta$ give the condition to be satisfied as $0\lt|x-a|\lt \delta\Rightarrow |f(x)-f(a)|\lt\epsilon$, thus excluding $x=a$ from consideration. – Arturo Magidin Feb 14 '12 at 22:25
  • @D.Thomine By the way, perhaps a translation might be "punctured limit", in analogy to the fact that if $U$ is a neighborhood of $p$, then $U-{p}$ is called a "punctured neighborhood of $p$". – Arturo Magidin Feb 15 '12 at 05:32

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This question has been answered in comments:

If $p$ is an isolated point, then we can always find $U$ such that $U∩Ω−p=∅$, in which case the condition $f(U∩Ω−p)⊆V$ is trivially satisfied. This means that $\lim_{x→p}f(x)=L$ for all $L$, which makes the notion of "limit" as an isolated point rather useless. (Intuitively: the limit asks what the function $f$ is doing "near", but not at the point $p$; if $p$ is isolated, then there is no "near" there).

As to your last question: doesn't matter: because $A⊆Ω$ is open with respect to the topology on $Ω$ if and only if there is a set $V$ that is open in $X$ such that $V∩Ω=A$. Since we are only considering what happens inside $Ω$, whether you look at $A$ or at $V$, you get the same intersection. – Arturo Magidin Feb 14 '12 at 21:50

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