We prove that the set F of all bounded functionally connected functions is boundary in the space of all bounded Darboux functions (with the metric of uniform convergence). Next we prove that the set of bounded upper (lower)semi-continuous Darboux functions and the set of all bounded quasi-continuous functionally connected functions is porous at each point of the space F.

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In the paper we give and investigate the basic properties of a method for improving continuity, quasi-continuity and the Darboux property of real functions defined on a metric Baire space. This "improvement" is carried out with the use of Blumberg sets.A. Katafiasz in her doctoral dissertation [5] (Also see [6] and [7].) introduced the notion of an α-improvable discontinuous function. The main idea of this work was to examine the possibility of the "removal" of the discontinuity of real functions defined on some subsets of the line. The author suggested a certain method for removing the discontinuity and investigated the structure of functions for which this method is efficient as well as the successive steps of the procedure of removing of the discontinuity. This dissertation has inspired our team's investigations.In our paper we give another method for "improving functions". In addition we show that our method is efficient for a considerably wider class of functions than A. Katafiasz's method, and that it may also be applied, in some cases, to the improving of quasi-continuity and the Darboux property. An additional merit of our method is the fact that the "improvement" is done in one step.