Can you explain me the definition of a boundary point ?
The definition is :
Let $A \subset \mathbb{R}^{n}$. A point $x \in \mathbb{R}^{n}$ is called a boundary point of $A$ if every neighborhood of $x$ contains at least one point in $A$ and a least one point not in $A$.
I attach a draw. 
Following the definition all the pictures are correctly but in my opinion I think that the last picture is OK when the boundary point is laying on that edge.
Another example, let's consider the following example :
Let $A=(a,b) \subset \mathbb{R}$. Then the boundary points of $A$ consists of te points $a$ and $b$. Why only the points $a$ and $b$? Why not the point $b+1$ or $a+1$ ?
Thanks :)
