n-point sets and graphs of functions

We prove that, for every n⩾2, there exists an n-point set (a plane set which hits every line in exactly n points) that is homeomorphic to the graph of a function from R to R; for n⩾4, there exist both 0-dimensional and 1-dimensional examples. This raises the question (which we do not answer) of whether n-point sets for different n's could be homeomorphic.