The graph of y = sin(1/x), in particular that it oscillates with increasing frequency as x gets closer to 0, has everything to do with the boundary.
Quote by filter54321
F is a squiggly line in R2. For every point in F (every point on the squiggly line) an open ball about that point will contain point both in F and in the complement of F. Therefore F is it's own boundary.

The problem is right here. All you've shown is that F is a subset of its boundary. There may be other points of R2 in the boundary.