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Let $(a\,;b)$ be an open interval of real numbers.

Let $L$ be some real number.

Iff ${\forall}\epsilon \in \mathbb R_{>0}:\exists\delta \in \mathbb R_{>0}:\forall x \in (a\,;b):0<|c-x|<\delta\implies |f(x)-L|<\epsilon$ then we say that $L=\displaystyle \lim_{x\rightarrow c}f(x)$.

So far I know that $c$ is some constant real number. I've also heard (not from every source though) that it needs to be a limit point of $(a\,;b)$. Is this a tautology, or are there cases where the proposition defining $L$ above is true, but $c$ isn't a limit point, therefore (?) making $L$ not a limit in those cases?

Is saying that $c$ represents a constant real number enough, or is this definition of $c$ incomplete?

asdasdf
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  • I don't think the first colon should be there (i.e. for all $\epsilon > 0$, $\exists \delta > 0$ etc...) and neither should the third – FraGrechi Feb 19 '17 at 20:08
  • Also, possible duplicate: http://math.stackexchange.com/questions/1476371/epsilon-delta-definition-of-a-limit-smaller-epsilon-implies-smaller ? – FraGrechi Feb 19 '17 at 20:14
  • $c$ is the point at which the limit of $f$ is being taking. Typically you'll want $c$ to be a limit point of the domain of $f$, because you are trying to analyze the behavior of $f(x)$ as $x$ approaches $c$ – joeb Feb 19 '17 at 20:14
  • Consider adding the following to the beginning of your definition. $f$ is continuous at $c \in (a, b)$ iff ... Does that help? – Homotopy Feb 19 '17 at 20:30
  • @GrancescoFrechi The colons don't have any meaning, they are supposed to separate the symbols to make the sentence "cleaner". I've seen this used fairly widely. – asdasdf Feb 19 '17 at 20:53
  • @joeb Does it need to be a limit point though? – asdasdf Feb 19 '17 at 20:53
  • @stableMatch This is about limit, not continuity. At any rate, I would think that making $c$ an interior point of $(a,b)$ is most natural the way the formula is written. – Arthur Feb 19 '17 at 21:05
  • @asdasdfsss Understood; you should be careful though, many texts treat colons as they do vertical lines (i.e. they take both to mean "such that"). When working with $\epsilon - \delta$ proofs, logical flow is very important – FraGrechi Feb 19 '17 at 23:29

1 Answers1

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It seems that you are confusing the notions of limit point $c$ of a set ( as the interval $(a,b)$) and of limit of a function when $x\to c$.

The first makes rigorous the intuition that, in the interval $(a,b)$ there are point whose distance from $c$ is small as we want .

The second says us what is (if it exist) the value of a function whan its argument is near the limit point $c$ and this value is called the **''limit'' for $x \to c**$.

But: $c$ is not the limit of the function.

Emilio Novati
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