Fuzzy approximate continuity

Approximately continuous functions were first introduced in connection with problems of differentiation and integration. The main idea of the approximate continuity of a function f is that the continuity conditions are true not necessarily everywhere but only almost everywhere with respect to some measure, e.g., Borel measure or Lebesgue measure. At the same time, it is known that functions that come from real life sources, such as measurement and computation, do not allow, in a general case, to test whether they are continuous or even approximately continuous in the strict mathematical sense. To overcome these limitations, fuzzy approximate continuity of functions is introduced and studied in this paper.