Transitivity of interval and fuzzy-interval extensions of interval functions

Let I = [a, b] be a real compact interval and f : I → I a continuous function. Define Kc([a, b]) the class of all non empty compact subinterval of [a, b] and let f̄ the natural extension of f to Kc([a, b]), that is to say, f̄(J) = f(J) for all J ∈ I([a, b]). Also, let Fc([a, b]) the class of all fuzzy-intervals with support contained in [a, b] and consider f̂ the Zadeh’s extension of f to Fc([a, b]). The aim of this paper is to show that f̄ and f̂ are not transitive for any arbitrary self continuous function f defined on I.