Why in the ε-δ definition of a limit is x∈D instead of just ℝ

Question

ε-δ definition:

$(\forall \epsilon >0) \ (\exists \delta >0)(\forall x \in D) (0 < \lvert x - a \rvert < \delta \Rightarrow \lvert f(x) - L \rvert < \epsilon)$

I don't understand why in the epsilon-delta limit definition x is defined as being x∈D instead of just using x∈ $\mathbb{R}$. Would specific issues arise for certain functions or types of functions if I were to use x∈ $\mathbb{R}$ instead?

Answer 1

I think that $D$ is the domain of the function. Take for example the function $f(x)=\ln(x)$.

You know that this function is a map between $\underbrace{(0,\infty)}_D\to\mathbb R$, so in this case it does not make sense to study continuity at a point $x = -2$ ,for example, as this function is not defined here. If you have a function $f: \mathbb R \to \mathbb R$, for example $f(x)=x$, then your $D = \mathbb R$ and then you can let $x \in \mathbb R$.

Answer 2

I would like suggest as answer conception of limit with respect to some set D in metrical spaces, for example Rudin W. - Principles of mathematical analysis,1976, 83-84p. :

4.1 Definition. Let $X$ and $Y$ be metric spaces; suppose $D \subset X$, $f$ maps $D$ into $У$, and $p$ is a limit point of $D$. We write $f(x) \to q$ as $x \to p$, or $$\lim\limits_{x \to p}f(x)=q$$ if there is a point $q\in Y$ with the following property: For every $\varepsilon >0$ there exists a $\delta > 0$ such that $d_Y(f(x),q)<\varepsilon$ for all points $\forall x \in D$ for which $0<d_X(x,p)<\delta$. The symbols $d_X$ and $d_Y$ refer to the distances in $X$ and $Y$, respectively.

If $X$ and/or $Y$ replaced by the real line, the complex plane, or euclidean space, the distances are replaced by absolute values or by norms.

Addition.(Tuesday, July 18 2023, for anonymous downvoter, who possibly have arguments for showed critic)

Formally limit for function $\lim\limits_{x\to c}f(x)=L$ with respect to $E$ is following (Vladimir A. Zorich - Mathematical Analysis I-Springer (2016), page 106 or Limit_of_a_function) :

$$(\forall \varepsilon>0)(\exists \delta>0)(\color{red}\forall x \in E)\Big(0<|x-c|<\delta\Rightarrow |f(x)-A|<\varepsilon \Big)$$

Also we can write it as following

$$(\forall \varepsilon>0)(\exists \delta>0)\Big(\color{red}\forall x \in E\cap \{x:0<|x-c|<\delta\}\Big)\Big( |f(x)-A|<\varepsilon \Big)$$

By choosing a different $E$ we can control the extent to which the second universal quantifier (indicated in red) is applied.

Let's consider the well-known Dirichlet function

$$D(x)=\begin{cases} 1, & x\in \mathbb{Q} \\ 0, & x\notin \mathbb{Q} \end{cases}$$

If we take $E= \mathbb{R}$, then limit doesn't exists, but it exists if $E= \mathbb{Q}$ or $E= \mathbb{R}\setminus\mathbb{Q}$.