In "Principles of mathematical analysis" Walter Rudin gives the definition of
$$\lim_{x \rightarrow p} f(x)=q$$
for $f: X \supset E \rightarrow Y$ with $X,Y$ metric spaces, in this way:
for every $\epsilon>0$ there exists a $\delta >0$ such that
$$d_Y(f(x),q)<\epsilon$$
for all points $x \in E$ for which
$$0<d_X(x,p)<\delta$$
where $d_X$ and $d_Y$ are the distances in $X$ and $Y$. After few pages he gives the definition of continuity:
$f$ is said to be continuous at $p$ if for every $\epsilon>0$ there
exists a $\delta >0$ such that$$d_Y(f(x),f(p))<\epsilon$$
for all points $x \in E$ for which
$$d_X(x,p)<\delta$$
My question is, why did he drop the $0<\dots$ in $d_X(x,p)<\delta$? I though that continuity in $p$ means that the limit of the function in $p$ exists and is equal to $f(p)$, so one should simply replace $q$ with $f(p)$ in the above definition of limit, why changing the condition on $x$?
Doing so a function like
$$f: \{1\} \rightarrow \mathbb{R}$$ which maps 1 to 5, for example, would be continous in 1, since $x=p$ is allowed, but what is the limit of $f$ in 1? It doesn't exist, right?
Is there a deep reason for such a counterintuitive definition?
EDIT:
Maybe I should better clarify my question. I know that this definition works, I am asking why considering a function continuos at isolated points (which is an immediate consequence of this defintion —> It turned out I was wrong about this).
The only reason I can think of is that this definition agrees with the topological one for continuity, but it doesn't seem to me a good reason since topology came after (and would have agreed no matter the convention).
Best Answer
The real question is, why is the $0<d_X(x,p)$ condition in the definition of the limit?
Basically, it doesn't hurt to allow $x=p$ in the continuity example, because (1) we know that $p\in E$, and (2), when $x=p$, $d_Y(f(x),f(p))=0<\epsilon$, so there is no reason to leave it out.
Essentially, in the continuity case, the case $x=p$ is trivially true.
On the other hand, when $p\in E$ in the limit definition, we don't want the limit to depend on $f(p)$, but only on the values $F(x)$ when $x\neq p$.
For example, when $f(x)=0$ for $x\neq p$ and $f(p)=1$, we want $\lim_{x\to p} f(x) = 0$, but that would not be true if we didn't have the condition $0<d_X(x,p)$ - without that condition, the limit is undefined.
Isolated points:
We consider isolated points to be points of continuity because we want in general that if:
$f:E\to Y$ is continuous at $p$ and $p\in E'\subset E$ then $f_{|E'}:E'\to Y$ to also be continuous at $p$.
However, note that the continuity definition could have said $0<D_X(x,p)$. That doesn't affect the continuity at $p$ one bit, so if you wanted to define all isolated points as points of discontinuity, you'd want some other definition completely.