@flawr: Something I'm thinking about: Voronoi [dis]continuities. Let's say that we are given a 2D curve in the plane. Each point in the plane is closest to some point on the curve. Informally, a point in the plane is Voronoi discontinuous if you need only move a little bit in a certain direction for the closest point on the curve to jump from one spot to another, without transitioning smoothly in between.
To put it another way, let's say the plane is the preimage and the curve is the image, where the mapping is defined by this Voronoi-style relation. Then a point is Voronoi-discontinuous i…