Definition of a limit between functions in metric spaces: Rudin vs Amann-Escher - excluding a limit point Amann-Escher in Analysis I define a limit of a function between metric spaces as follows:

Let $X,Y$ be metric spaces, $D \subseteq X$ and $a \in X$ a limit point of $D$. Let $f\colon D\to Y$ be a function. We write $\lim_{x\to a} f(x) = y$ if for any sequence $(x_n)$ in $D$ which converges to $a$ in $X$, the sequence $(f(x_n))$ converges to $y$ in $Y$.

They then prove that this is equivalent to the following:

For each neighborhood $V$ of $y$ in $Y$, there is a neighborhood $U$ of $a$ such that $f(U\cap D) \subseteq V$.

Rudin's definitions are slightly different:

Let $X,Y$ be metric spaces, $D \subseteq X$ and $a \in X$ a limit point of $D$. Let $f\colon D\to Y$ be a function.
  We write $\lim_{x\to a} f(x) = y$ if for any $\epsilon > 0$ there exists $\delta > 0$ such that $d_Y(f(x),y) < \epsilon$ for all $x \in D$ such that $d_X(x,a) < \delta$. 

As far as Amann-Escher definition goes, this would be equivalent to the following:

For every neighborhood $V$ of $y$ in $Y$, there is a neighborhood $U$ of $a$ such that $f(U\cap (D\setminus\{a\})) \subseteq V$.

For the sequential version of a limit definition, Rudin requires that $(f(x_n))$ converges to $y$ not for all $(x_n)$ in $D$ which converge to $a$, but for all $(x_n)$ in $D\setminus\{a\}$ which converge to $a$.
I understand that Rudin's version is standard. But what is really wrong with Amann-Escher's version that it can't be used? Why do we need to exclude $a$?

In Amann -Escher definition if $x_n \to a$ (and $a \in D$) then $(x_1,a,x_2,a,x_3,a...)$ also converges to $a$. So this forces the limit of $f(x_n)$ to be $f(a)$. So this becomes definition of continuity at $a$ rather than existence of the limit. 
