Approximately Continuous Transformations

1. An interesting class of real functions of a single real variable, the approximately continuous functions, was introduced by Denjoy [l] in his work on derivatives. The two principal facts discovered by Denjoy are that these functions are of Baire class 1 and have the Darboux property. Ridder [2] showed that the arguments of Denjoy apply to real functions of n variables. In this paper we discuss approximately continuous transformations from euclidean spaces into arbitrary metric spaces. We show that the image under such a transformation is always separable, that the transformation is of Baire class 1, and that it has a Darboux property. In §2 the Darboux property for real functions of a real variable is discussed. In §3 the notions of approximate continuity and metric density are defined in our context. In §4 it is shown that the image of En under an approximately continuous transformation is separable and such transformations are of Baire class 1. In §5 we introduce the notion of homogeneity of sets relative to metric density. En may be topologized by taking the homogeneous sets as open sets.1 We show that the open connected subsets of En are connected in this topology. The approximately continuous transformations are the continuous transformations in the new topology. In §6, we define ¿-regular sets, closed sets with connected interior and certain boundary restrictions. It is shown that approximately continuous transformations take irregular sets into connected sets, i.e., they have a Darboux property. Further remarks on connectivity are made in §7.