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 nʹs could be homeomorphic.