Definitions of the 'limit points' I knew that the definition of limit points $x$ of a subset $E$ of $\mathbf{R}^d$ is :

For any open ball $B$ containing $x$, $(B\setminus \{x\})\cap E\ne\emptyset$

Or something like these arguments containing 'except the point $x$'.
But in the Real analysis textbook (page 3), by STEIN & SHAKARCHI,  I found the definition of limit points slightly different : 

A point $x\in\mathbf{R}^d$ is a limit point of the set $E$ if for every $r>0$, the ball $B_r(x)$ contains points of $E$.

I am convinced that the above two definitions are not equivalent to each other.
Thus I wonder what are the key differences when using the second definition of STEIN comparing with the first one.
THANKS.
 A: I think that by "points" they mean more than one, in which case the definitions would be equivalent. Otherwise if you take $E$ to be a singleton, say $E=\{a\}$, then $a$ would be a limit point, which would not be very consistent with the name... We use limits points in the definition of limits. With Stein's definition, a lot of statements about limits would be empty. In the definition of $ \lim_{x\to a}f(x)$ the point $a$ is a limit point of the domain $E$ of $f$ and you never take $x=a$ and so there is not much to check if $E=\{a\}$.
You would also lose the important property that if $a$ is a limit point of $E$, then you can find a sequence $x_n\in E$ of distinct points such that $x_n\to a$.
It would not affect properties like " a set is closed iff it contains all its limit points". 
Overall, this definition means that they are adding isolated points to our usual definition of limit points. I don't like it very much, but I think that most theorems would continue to work, with small exceptions here and there for isolated points.
A: The second definition differs from the standard definition. In fact, with that definition every point $x \in E$ is a limit point of $E$. We typically don't want that. Instead it should read

A point $x \in \mathbb R^d$ is a limit point of the set $E$ if for
  every $r > 0$, the ball $B_r(x)$ contains a point in $E \setminus \{x\}$.

This definition is in fact equivalent to the first definition in your post:
Let $x$ be a limit point according to my definition. If $B$ is an open ball containing $x$ then there is some $r > 0$ such that $x \in B_r(x) \subseteq B$. Hence there is some $y$ such that 
$$y \in (B_r(x) \setminus \{x \}) \cap E \subseteq (B \setminus \{x \}) \cap E.$$
So $x$ is a limit point of $E$ according to your first definition.
The converse is trivial as we may, for any $r > 0$, take $B := B_r(x)$.
A: 
A point $x\in\mathbf{R}^d$ is a limit point of the set $E$ if for every $r>0$, the ball $B_r(x)$ contains points of $E$.

The above definition doesn't allows the intuitive idea that a point $x$ is a limit point of a set $E$ means that there are infinitely many points of $E$ that are arbitrarily "close" to the point $x$. However I do not know which came first, the intuitive idea or the definition. Most of the time limit points are called accumulation points or cluster points. Those terms back up the intuitive idea behind such concepts.
In Engelking's text General Topology the term accumulation point is used and he goes on to define it in the usual sense; 

A point $x$ in a topological space $X$ is called an accumulation point of a subset $E$ of $X$ if $x\in\overline{E\setminus \{x\}}$. 

So I think it is possible that these terms aren't synonymous in Stein's context.  
