The Devil's staircase function has derivative zero on the complement of this set but is not constant.
This space is the universal compact metric space in the sense that there is a continuous surjection from it to any other non-empty compact metric space. A "wild" example of this space which is embedded in R^3 [read "R three"] with its complement not simply connected is Antoine's necklace. Though it has measure zero, analogues with positive measure that are still nowhere dense can be constructed as so-called "fat" examples. The Devil's staircase function has derivative zero on the complement of this set but is not constant. By the Moore-Kline theorem every non-empty, compact, perfect, and totally disconnected metric space is homeomorphic to it. A multi-dimensional version is called this type of dust. For 10 points, name this set, constructed by starting with the unit interval and then successively removing the middle-thirds of all intervals in sight.
This space is the universal compact metric space in the sense that there is a continuous surjection from it to any other non-empty compact metric space.
{Cantor} set or {Cantor} space or {Cantor} dust