MODERN ASPECTS OF QUASICONTINUOUS FUNCTIONS

Abstract

The concept of continuity became an inspiration for the creation of topology. In the 20th century, there were many attempts to generalize this concept. One such attempt was made by the Polish mathematician Stefan Kempisty in the 1920s, when he introduced the concept of quasicontinuity. Various concepts of continuity in topological spaces are defined and their interconnections are examined. In this paper, we consider only Kempisty's quasicontinuity for functions of several real variables, in order to give a new approach to these functions regarding various applications in many fields of technical sciences. New proofs are provided for three of his fundamental theorems from his original work. A new generalization of Kempisty's theorem is made with three new theorems, each of which is individually stronger than Kempisty's theorem and which address the issues posed by his theorem. Two of these theorems, in general form, were proved by J. Ewert, who obtained them while examining multiplied functions, but the proofs given in this paper do not utilize the mentioned results of J. Ewert.