On Continuity and Monotonicityof Darboux

In this paper we consider the problems connected with the continuity of Darboux transformations and the monotonicity of the restrictions of these transformations. We show that it becomes possible to give answers to many questions concerning these problems if our considerations are connned to the family of c-functions which is deened in the paper. In paper DG] (1875), the rst example of a discontinuous Darboux function was given. Since then, there have appeared many papers devoted to the studies of the properties of these functions. The proving of a series of interesting properties for real Darboux transformations of a real variable became a cause of the search for a generalization of the notion of a Darboux function to the case of transformations deened and taking their values in more general spaces. Diierent ways of the generalizations can be found, among others, in papers BB], GK], PR], and the speciication and discussion of many of them | in paper JJ]. The deenitions presented below are analogous to those of Darboux(B) transformations and weakly connected ones considered in papers BB] and GK]. However, what diierentiates one from another is that we free ourselves from the strictly deened classes of sets, considered in these papers. So, let L be some family of connected sets (in the sequel, unless otherwise stated, L will always denote a xed family of connected sets) in a topological space X and let f : X ! Y. We say that f is a Darboux (L) function if f(C) is a connected set for any C 2 L. We say that f is a Darboux(L) function if f(C) is a connected set for any C 2 L. It will be convenient to our considerations to adopt the following deenition, too: We say that B is an L-base of the topological space X if B is an open base of