On real functions with graphs either connected or locally connected

Let S denote the family of all subspaces of the plane that are graphs of functions from the real line R to itself. We prove that S has two subfamilies G,H of spaces such that the cardinality of G is c (the cardinality of the continuum) and the cardinality of H is 2^c, every space in the family G is completely metrizable, each element of H is a dense subset of the plane and the elements of the union of G and H are pairwise non-embeddable (i.p. pairwise non-homeomorphic) subspaces of the plane. On the other hand, the family S contains precisely countably infinitely many locally connected spaces up to homeomorphism, and if X,Y are such spaces then X is embeddable into Y. Furthermore, if T is a topology on the set R finer than the Euclidean topology and the space (R,T) is separable and locally connected then the space is locally compact and homeomorphic to some space in S. In a very natural way we establish a complete classification of all these refinements T of the real line.