I am reading "Introduction to Analysis I" (in Japanese) by Mitsuo Sugiura.
This book is the standard and the most popular introductory analysis book in Japan.
The standard definition for limit:
Let $f$ be a function from $A\subset\mathbb{R}^n$ to $\mathbb{R}^m$ and $a\in\overline{A},b\in\mathbb{R}^m$.
We write $$\lim_{x\to a}f(x)=b$$ if for any positive real number $\varepsilon$, there exists a positive real number $\delta$ such that $$0<|x-a|<\delta\text{ and }x\in A\implies |f(x)-b|<\varepsilon.$$
Definition for limit in this book:
Let $f$ be a function from $A\subset\mathbb{R}^n$ to $\mathbb{R}^m$ and $a\in\overline{A},b\in\mathbb{R}^m$.
We write $$\lim_{x\to a}f(x)=b$$ if for any positive real number $\varepsilon$, there exists a positive real number $\delta$ such that $$|x-a|<\delta\text{ and }x\in A\implies |f(x)-b|<\varepsilon.$$
Definition 2 for limit in this book:
Let $f$ be a function from $A\subset\mathbb{R}^n$ to $\mathbb{R}^m$ and $b\in\mathbb{R}^m$.
Let $B\subset A$ and $a\in\overline{B}$.
We write $$\lim_{x\to a\\x\in B}f(x)=b$$ if for any positive real number $\varepsilon$, there exists a positive real number $\delta$ such that $$|x-a|<\delta\text{ and }x\in B\implies |f(x)-b|<\varepsilon.$$
Let $f$ be a function from $A\subset\mathbb{R}^n$ to $\mathbb{R}^m$ and $b\in\mathbb{R}^m$.
Let $a\in A$ and $B:=\{x\in A\mid x\neq a\}$ and $a\in\overline{B}$.
Then $$\lim_{x\to a\\x\in B} f(x)=b$$ is equivalent to the standard definition for limit.
I think the definition for limit in this book is not standard.
What are the advantages and disadvantages of this definition for limit?
Are there any other books which adopt this definition for limit?
