Is it true that, given a set E, every point in the set of boundary points of E is an accumulation point of the set of boundary points?

Question

Suppose $E\subset \mathbb{R}^n.$ Is it true that for all $x\in\partial E,$ $x$ is an accumulation point of $\partial E$?

The reason I think this is true (despite my feeling it is false) is that we have $$\partial E=\{ p \in \mathbb{R}^n, \forall r>0 \text{ one has } B(p,r)\cap E\neq\emptyset \text{ and } B(p,r)\cap E^C \neq \emptyset \}$$

which implies, with some extra reasoning, that every ball about every point in $\partial E$ Intersects $S-\{p\}.$

Answer 1

$E=(0,1) \subseteq \mathbb R$ only has two boundary points and these are not accumulation points of that finite boundary....