If $A$ is a non-empty subset of $R$ is every point of $A$ either an isolated point or a limit point? From what I can gather, a limit point is a point $a \in A$ such that the open ball $B(a; \delta)$ contains a point other than $a$. While an isolated point is a point $b \in A$ such that the open ball $B(b;\delta)$ contains no point different from $b$. So, is every point always a limit point or an isolated point?
 A: You should be a lot more careful with your definitions, there are a couple important parts missing: 


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*Let $X$ be a metric space and $A\subset X$.  A point $a\in X$ is said to be a limit point of $A$ if for all $\delta>0$, there is an $x\in A\cap B(a;\delta)\backslash\{a\}$.


Notice: a limit point need not be in $A$!  Example: $A=(0,1)\subset\Bbb{R}$.  Both $0$ and $1$ are limit points of $A$.


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*Let $X$ be a metric space and $A\subset X$.  A point $b\in A$ is an isolated point of $A$ if there exists $\delta>0$ such that $A\cap B(b;\delta)\backslash\{b\}=\emptyset$.


Notice: isolated points must be in $A$.
So to answer your question, it is easy to show that $b\in A$ is an isolated point of $A$ if and only if $b$ is not a limit point of $A$.  So if you only consider points in $A$, then the answer to your question is yes.  However the statement "a point is either a limit point or an isolated point" is technically false, since it is possible to be neither if your point is not in $A$ (e.g. take $x=2$ with $A=(0,1)$; $x$ is neither a limit point nor an isolated point of $A$).
A: In your definition, you forgot the important part: the $\exists$ and the $\forall$. Once you clarify this, as @GEdgar said, the answer will clearly be yes.
A: A limit point of a set $B$ is a point $a$ such that EVERY open ball about $a$, no matter how small, contains some point in $B$ other than $a$ itself.  A limit point of $B$ need not be a member of $B$, but may be.  You can't omit the word "every".
An isolated point of $B$ is a point $a$ that is a member of $B$, but such that some open ball centered at $a$ contains no point of $B$ other than $a$.
A point $a$ that does not belong to $B$, and is the center of some open ball that does not intersect $B$, is neither a limit point of $B$ nor an isolated point of $B$.
