Free continuity

In this thesis, the notion of continuity defined by Weierstraß in the middle of the 19th century has been studied utilizing various generalizations of the subject today known as sequential continuity. It is well-known that in the real line if a function is continuous then it is continuous and the converse is also true, but in a general setting, this is not true. Many properties and generalizations of sequential continuity have been studied by various authors such as (Schoenberg, 1959; Srinivasan 1980; Antoni, Salat, 1980) and studied in (Spigel & Krupnik, 1994; Connor & Grosse-Erdman, 2003). But the most important problem they had was the so-called dichotomy of the new functions defined by some method being either linear or already continuous in the classical sense. A step for breaking this dichotomy has been considered in this thesis by restricting the scope of the universal quantifier and by using the notion of natural density. To do this, the notion of free real numbers is defined, and with the help of this notion, the class of free continuous functions is defined. It has been shown that while preserving the class of all classical functions the class of free continuous functions cannot be compared with the class of almost continuous functions and the class of functions such that the absolute value of functions is Lebesgue integrable (𝐿1 space).