Fuzzy Continuity of Almost Linear Operators

In this paper, the author studies relations between fuzzy continuity and boundedness of approximately linear operators in the context of neoclassical analysis. The main result of this paper (Theorem 1) demonstrates that for approximately linear operators, fuzzy continuity is equivalent to boundedness when the continuity defect (or measure of discontinuity) is sufficiently small. The classical result that describes continuity of linear operators becomes a direct corollary of this theorem. Applying Theorem 1, we demonstrate (Theorem 2) that for linear operators in normed vector spaces, fuzzy continuity coincides with continuity when the continuity defect is sufficiently small, i.e., when it is less than one. Results are oriented at applications in physics, theory of information and other fields where operator equations play an important role. Several open problems and directions for future research are considered at the end of the paper.