Boundary point

A point where every ball, no matter how small, catches points both inside and outside the set.

A point $\mathbf{a}$ is a boundary point of a set $S$ if every ball $B_\varepsilon(\mathbf{a})$ , for every $\varepsilon>0$ , contains points of $S$ and points not in $S$ . A boundary point need not itself belong to $S$ .

Examples

Every point on the unit circle $x^2+y^2=1$ is a boundary point of the open disk $x^2+y^2<1$ , even though none of them belong to it.