A note on somewhat fuzzy continuity

TL;DR

This paper clarifies the relationships among various types of fuzzy functions, showing equivalences and hierarchies, and refines understanding of fuzzy continuity concepts in fuzzy topology.

Contribution

It proves the equivalence of somewhat fuzzy continuous and semiopen functions and establishes that somewhat fuzzy continuous functions are weaker than fuzzy semicontinuous functions.

Findings

Somewhat fuzzy continuous and semiopen functions are equivalent.

Somewhat fuzzy continuous functions are weaker than fuzzy semicontinuous functions.

Clarifies the hierarchy of fuzzy function types in fuzzy topology.

Abstract

Thangaraj and Balasubramanian introduced the so-called somewhat fuzzy semicontinuous and somewhat fuzzy semiopen functions. Two years later, the same authors defined two other types of functions called somewhat fuzzy continuous and somewhat fuzzy open without indicating connections between them. At first glance, we may easily conclude (from their definitions) that every somewhat fuzzy continuous (resp. open) function is somewhat fuzzy semicontinuous (resp. semiopen) but not conversely. In this note, we show that they are equivalent. We further prove that somewhat fuzzy continuous functions are weaker than fuzzy semicontinuous functions.