Connectedness properties of special subsets of the plane

In this paper we investigate the connectedness properties of graphs of functions from the real line R to itself. We show, among other things, that (i) a totally disconnected graph of a function from R to R need not be zero-dimensional, (ii) the graph of every injection from R to R is zero-dimensional provided that it does not contain arcs. As a corollary of the second result we obtain an easy proof of the well-known fact that every two-point set is zero-dimensional. We conclude the paper by showing that there is a two-point set which is not homeomorphic to the graph of a function from R to R.