# What's $c$ in the $(\epsilon, \delta)$-definition of limit?

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?

• 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 Feb 19, 2017 at 20:08
• Also, possible duplicate: math.stackexchange.com/questions/1476371/… ? Feb 19, 2017 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, 2017 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? Feb 19, 2017 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. Feb 19, 2017 at 20:53

## 1 Answer

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.