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In Rudin's "Principles of Mathematical Analysis", Definition 4.33 (p98) states:

Definition 4.33 $\quad$ Let $f$ be a real function defined on $E.$ We say that $f(t)\to A$ as $t\to x$, where $A$ and $x$ are in the extended real number system, if for every neighborhood $U$ of $A$ there is a neighborhood $V$ of $x$ such that $V \cap E$ is not empty, and such that $f(t)\in U$ for all $t\in V\cap E, t\ne x.$

I'm a little confused however by the remark right beneath this definition:

A moment's consideration will show that this coincides with Definition 4.1 when $A$ and $x$ are real.

My question is actually three-fold:

1) In Definition 4.1 (quoted below), it is explicitly stated that $x$ ($p$) has to be a limit point of $E$. Can this be inferred from Definition 4.33? It appears to me that according to Definition 4.33 the limit of a function can also be defined even if $x$ is an isolated point of $E.$ For example, let $E=\{0\}\cup \{1\} \cup [3,4]$, and $f(0)=f(1)=0, f(t)=t, t\in [3,4]$. Then with $V$ being a neighborhood of radius $2$ around $0$, we have $f(t)\to 0$, as $t\to 0,$ don't we?

2) Similarly, for limits at infinity, i.e. $x=\infty$, Definition 4.33 doesn't require the domain $E$ of $f$ to be unbounded, does it? For example, if $f(x)=1$ for $x\in[0,1]$, then we may also say $f(x)\to 1$ as $x\to \infty$? (e.g. with $V=(0.5, \infty)$ we clearly satisfy Definition 4.33.)

3) Should we or should we not add these assumptions (or implicitly assume them) when using Definition 4.33? (i.e. $x$ has to be limit point of $E$, or $E$ is unbounded above or below.)

Thanks a lot!

Definition 4.1 $\quad$ Let $X$ and $Y$ be metric space; suppose $E \subset X$, $f$ maps $E$ into $Y$, and $p$ is a limit point of $E$. We write $f(x)\to q$ as $x\to p$, or $\lim_{x\to p}f(x)=q$ if there is a point $q \in Y$ with the following property: For every $\epsilon > 0$ there exists a $\delta > 0$ such that $d_Y(f(x),q)<\epsilon$ for all points $x\in E$ for which $0<d_X(x,p)<\delta$.

syeh_106
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    Your understanding is correct but there is no need to add extra considerations for special cases in the statement 4.33. It assumes that know that in $[-\infty,\infty],$ a nbhd of $\infty$ is any $S$ such that $S\supset (r,\infty)\cup {\infty}$ for some real number $r,$ and similarly for nbhds of $-\infty.$ – DanielWainfleet Jan 18 '17 at 03:28
  • @user254665 Thanks for the comments! Just to clarify a bit more... About limits at infinity, my question is more about whether it makes sense (or is it conventional) to write $f(x)\to A$ as $x\to \infty$, when the domain of $f$ is bounded? – syeh_106 Jan 18 '17 at 03:44
  • Yes it's common ,e.g. $2^{-x}\to 0$ as $x\to +\infty.$ – DanielWainfleet Jan 18 '17 at 04:15
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    But if you are restricting $2^{-x}$ to the interval $[0,1]$ then does it really make sense to talk about $x\to\infty$? That is the question here, no? – Jürgen Sukumaran Mar 05 '20 at 14:34
  • @TSF Yes. Hence my questions #2 and #3. It appears that Def 4.33 does not reduce to Def 4.1 and $A$ and $x$ are real. For the example in #2, according to Def 4.33, we may say $f(x)\to 1$ as $x\to \infty$, couldn't we? It doesn't make sense, but Def 4.33 allows it. – syeh_106 Mar 07 '20 at 15:44
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    I think the point in question should have to be a limit point of $E$ but I'm not sure if that's what Rudin meant or not. – Jürgen Sukumaran Mar 09 '20 at 09:25
  • I have the same question +1. So the conclusion is that both the definitions are "similar" but not the same because in def. 4.33, limit may exist for an isolated point also which is not the case with def. 4.1. Am I correct? – Koro May 02 '21 at 12:18
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    See https://math.stackexchange.com/questions/1365208/definition-of-the-limit-of-a-function-for-the-extended-reals (which is answered in the comments). – Hans Lundmark Apr 14 '23 at 17:56

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