The function y
= sin (1 / x
It goes up and down like an ordinary sine wave, within the same bounds y = 1 and y = −1, but the peaks and troughs spread out to infinite rarefaction as x approaches infinity, and bunch up to infinite thickness as x approaches 0.
It is symmetrical across the y-axis and is undefined at x = 0.
The topologically interesting behaviour is because it asymptotes to all of the y-axis between −1 and 1 as it nears 0 from either side. It tends towards a space-filling curve without actually being one.
If you take the union of the graph with the limit segment at x = 0, you get a topological space which has the comparatively unusual property of being connected but not path-connected. That is, although the limit doesn't touch the curve, it's arbitrarily close to it and they can't be put into separate non-intersecting regions. But you can't actually get from one to another by a continuous path.