limit of a function defintion

Question

Definition(limit of a function):

$f:D \subset \mathbb{R}^n \to \mathbb{R}^m$, $x_0 \in \mathbb{R}^n$ limit point of $D$

Then $ \lim_{x \to x_0}f(x)=y$ means that for each sequence $x_n \in D\setminus \{x_0\}$ with $\lim_{n \to \infty} x_n =x_0$ there holds $$\lim_{n \to \infty}f(x_n)=y.$$

my question: Assumed that $x_0 \in D$, why do we require that our sequence $x_n$ is in $D\setminus \{x_0\}$ and not in $D$? Does it even matter?

Answer 1

You forgot to add to your definition that $\lim_{n\to\infty}x_n=x_0$.

Anyway, the idea is that the value of $f$ at $x_0$ does not matter. So that, for instance, if you define$$\begin{array}{rccc}f\colon&\mathbb R&\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}0&\text{ if }x\neq0\\1&\text{ if }x=0,\end{cases}\end{array}$$then $\lim_{x\to0}f(x)=0$.