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If I understand correctly, most definitions of 'limits' require that the function either a) be defined in an open neighborhood around the relevant point or b) more permissively, that the relevant point is a limit point; the definition of 'continuity' is then given a special case so that functions are continuous at isolated points. Why not extend the notion of 'limit' so that the limit of a function at an isolated point is just whatever the function's value is there? Is there some good reason not to?

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    Functions are _always_ defined in open neighborhoods around a point, if "open neighborhood" is defined correctly (relative to the correct subspace).2011-03-16
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    What about discontinuous functions? Taking the step function, the one-sided limit is not equal to the function's value at that point.2011-03-16

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You could make that definition I suppose, but what use would it have, and how would it relate to the usual notion of limit?

Let's look at what a limit of a function $f$ at a point $x$ should mean (let's say for a real-valued function on a metric space). You want the limit of $f$ at $x$ to be a real number $L$ such that for all $\varepsilon>0$ there exists a $\delta>0$ such that $0

For one of the applications of limits, namely continuity, not defining the limit at an isolated point causes no problem. You can say that $f$ is continuous at a point $x$ in the domain if for all $\varepsilon>0$ there exists a $\delta>0$ such that $d(x,y)<\delta$ implies $|f(y)-f(x)|<\varepsilon$. If $x$ is an isolated point, then this will always be true, because for sufficiently small $\delta$ the only $y$ with $d(x,y)<\delta$ is $x$.

Added: I was writing when Alex posted, and part of my post makes a similar point to his. Qiaochu's comment on Alex's post gives an answer to my question at the beginning of my post. Making this definition allows continuity to be defined in terms of respecting limits without making isolated points a special case, something I had overlooked.

Nonetheless, continuity can be defined in terms of respecting limits without actually defining the limit of a function at a point. A function $f$ between metric spaces [resp. topological spaces] is continuous at $x$ if for every sequence [resp. net] $(x_n)_n$ in the domain converging to $x$, $\lim_n f(x_n)=f(x)$. In case $x$ is an isolated point, a sequence converging to $x$ is eventually constantly equal to $x$, so this will be satisfied.

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    Maybe I'm missing something, but I don't agree: if there is no point $y$ sucht that $0$\delta$ *in the domain of $f$*, then $f(y)$ simply does not exist: $f$ is not defined at $y$. Hence the inequality $\mid f(y) - L \mid < \varepsilon$ is not vacuously true: there is no such inequality. Am I wrong? – 2011-03-16
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    Maybe this is more a problem with my wording. If no such $y$ exists, then a statement that involves such $y$ is vacuously true. Whether or not a hypothetical $f(y)$ even makes sense is not considered in that case. As usual with vacuous truth, I can consider what the negation of the statement would be and use the law of excluded middle: to negate, there would have to exist a $y$ such that $0$|f(y)-L|\geq \varepsilon$, but no such $y$ exists. But if I am wrong, please inform me! – 2011-03-16
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    I would have always thought that you cannot put a term that doesn't exist, like that $f(y)$, into an equation... :-? Moreover, to talk about the limit of a function at an isolated point -though maybe not extremely useful in elementary calculus-, makes perfectly sense: it's just the value of the function at this point. And this is not a definition, but a consequence of Rudin's definition -deleting the condition that $x$ must be a limit point.2011-03-16
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    @Agustí: What definition exactly do you consider this to be a consequence of? I don't have a particular text reference on hand to dispute this claim.2011-03-16
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    I'm reminded of an exercise in Royden's *Real analysis*: "Show that if $x\in\emptyset$, then $x$ is a green-eyed lion." Of course the point is to get a student better acquainted with the notion of vacuous truth. What exactly it would mean for an element of a set to be equal to a green-eyed lion need not enter the picture; it surely is false that there exists an element of $\emptyset$ that is not equal to a green-eyed lion.2011-03-16
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    My point is that $f(y)$ is *not* put into an equation or any other relation, because no $y$ exists in the first place. The negated statement is one that is either true or false: "There exists $\varepsilon>0$ such that for all $\delta>0$ there exists $y$ such that $0$|f(y)-L|\geq\varepsilon$." In this case, you can see that this will be false at an isolated point by showing that there is a small enough $\delta$ precluding the existence of $y$ even satisfying the first condition. – 2011-03-16
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    Then again, there is a good chance that I'm missing some subtlety in the meaning of vacuous truth. If that is the case, then I would be further convinced that directly trying to generalize the definition of limit of a function to an isolated point is not a useful idea. (Of course, this does not interfere in any way with the definition of continuity, either with $\varepsilon-\delta$s, or in terms of limits of sequences/nets.)2011-03-16
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    I have not heard the term "respecting limits" before. Does it mean "limits that are consistent with continuity definition"?2015-07-30
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    @FreshAir, I think it means $\lim_nf(x_n)=f(\lim_nx_n)$.2015-08-03
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    This reminds me of the sequence definition of limits of functions - if your statement is true for all sequences $x_n$ that converges to $a$, then that implies $\lim_{x\to a}f(x)=f(a)$, doesn't it?2015-08-03
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    @FreshAir: Yes if the domain is first countable, e.g., a metric space, e.g., a subspace of $\mathbb R^n$, and if $a$ is not an isolated point.2015-08-03
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    very useful post thank you.2017-08-09
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Okay, now that I've checked Rudin to see that this really is the definition he gives of a limit of a function, I have an answer, but I don't like it. The motivation behind the definition of $\lim_{x \to a} f(x)$ is that you want to understand what $f$ is doing in a neighborhood of $a$ in order to compare it to what is happening at at $a$. If $a$ is an isolated point, there's nothing to compare.

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    +1: I don't really like it either. Fortunately there is no such ambiguity for limits of sequences or nets, and when the notion of limits of functions is introduced in calculus, it is usually for functions defined in a neighborhood (or punctured neighborhood) about a point in $\mathbb{R}^n$. I think this is why I had overlooked the point you made in your first comment on Alex's answer.2011-03-16
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    @Jonas: yes, what really annoys me is that this definition is not equivalent to "the common value of $\lim_{n \to \infty} f(a_n)$ where $a_n \to a$."2011-03-16
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    Another possible complication with isolated points is that limits will always exist and always be non-unique. This is not terrible if you are used to limits in arbitrary topological spaces, but for those used to Hausdorff spaces, it's a bit of an issue...2011-03-16
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    Nice answer a good way of putting things.2017-08-09
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    @QiaochuYuan Using "the common value of $\lim_{n \to \infty} f(a_n)$ where $a_n \to a$" as the definition of $\lim_{x \to a} f(x)$ makes it so that $\lim_{x \to a} f(x) = f(a)$ when $a$ is isolated. This is nice because it agrees with the fact that $f$ is continuous at isolated points (see Definition 4.5 on page 85-86 of Rudin). However, this definition has a major defect: It fails to assign a unique value to limits like $\lim_{x \to 0} f(x)$ for functions like $f(0)=0$, $f(x)=1$ if $x \neq 0$. Note 0 is not isolated here.2018-01-22
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    @QiaochuYuan If you look at Theorem 4.2 on page 84 of Rudin you will see the correct form: "the common value of $\lim_{n \to \infty} f(a_n)$ where $a_n \to a$ and $a_n \neq a$." Of course, this only works when $a$ is a limit point.2018-01-22
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The entire point of the notion of a 'limit' is to capture the behavior of a function as it gets "close" to a point, which is not possible for isolated points, thus there is no utility to extending the notion in the way you described. Your definition would make "limit" the same as "value" at these points, which is not very useful.

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    Why isn't it useful? If you want the theorem "a function (say, between metric spaces) is continuous if and only if it respects limits" this won't make sense if the domain has isolated points unless you define what limits at an isolated point mean.2011-03-16
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    But that would only save you the trouble of saying "except at isolated points", while invalidating numerous theorems about limits such as "a function is continuous if and only if its value is equal to its limit at every point".2011-03-16
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    I don't understand your second point. Functions are always continuous at isolated points.2011-03-16
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    @Qiaochu: Yes, you're right. Sorry, I just whipped that out of my hat without thinking. But I feel it would invalidate statements about when limits can be evaluated, at least as they are presently phrased.2011-03-16
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    Aren't these two statements the same? 1. "a function (say, between metric spaces) is continuous if and only if it respects limits" 2. "a function is continuous if and only if its value is equal to its limit at every point"2015-07-30