Definition 4.33 of Rudin's Principles of Real Analysis:
Let $f$ be a real function defined on $E \subset R$. We say that $f(t) \rightarrow A$ as $t \rightarrow x$ where $A$ and $x$ are in the extended real number system, if for every neighborhood $U$ of $A$ there is a neighborhood $V$ of $x$ such that $V \cap E$ is not empty, and such that $f(t) \in U$ for all $t \in V \cap E$, $t \neq x$.
Rudin then goes on to say this definition coincides with the epsilon-delta definition for when $A$ and $x$ are confined to $\mathbb{R}$. However for the epsilon-delta definition $x$ must be a limit point of $E$, whereas for Definition 4.33 this need not be the case. For example:
Let $E \equiv \{1\}$ and $f(1) \equiv 5$. Then $f(t) \rightarrow 5$ as $t \rightarrow 2$ since, for any neighborhood $U$ of $5$, $V \equiv (0, 4)$ is a neighborhood of $2$ for which $V \cap E = \{1\} \neq \emptyset$ and $f(1) = 5 \in U$. Clearly $x = 2$ is not a limit point of $E = \{1\}$.
It seems like this confusion could be avoided if, like in the epsilon-delta definition, Definition 4.33 required $x$ to be a limit point of $E$.
My question is:
Is there a reason $x$ is not required to be a limit point in definition 4.33?
Edit: My feeling is that the definition is only useful if $x$ is a limit point of $E$ and so there is no point in stipulating it. Therefore when Rudin states the definitions coincide (for real values) he implicitly considers $x$ a limit point of $E$.
