Ah, so if my target set is (0, pi-3) and the demon feeds me the digits of pi-3, I will be rolling forever, but if the demon feeds me the digits of pi-3-epsilon (or any other number in (0, pi-3)) I will be able to stop after a finite amount of rolls.

I think the type of set that this defines is "Sets with boundary of measure zero" where the boundary is the closure minus the interior.

That sounds right to me, although I don't understand measure very well. I was informally thinking of this property as "continuousness".