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

Question

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?

Answer 1

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.