On some limit and continuity types

In this thesis, the limit concept that is one of the fundamental concepts of mathematical analysis is considered. By the definition of the clasical limit, at a point α∈R, the limit of a function f from R to R is L∈R if and only if there exists δ>0 such that the set {x∈(α-δ,α+δ)\{α}:|f(x)-L|≥ε} is empty for each ε>0. In this study, it is investigated what happens when the above set is not empty. In this way, new limit definitions are obtained. Theorems on some basic properties of these new limit definitions (existence, uniqueness, additivity, multiplicitivity, etc.) are obtained. In addition, the limit definitions we have obtained are compared with the previous limit definitions and some examples are given.