Approximate Fuzzy Continuity of Functions

Introduction

A natural definition of function continuity is based on the classical concept of sequence convergence. At the same time, there are more general concepts of sequence convergence. One of them is the concept of statistical convergence, introduced by Steinhaus (1951) and Fast (1951) and later reintroduced by Schoenberg (1959). Statistical convergence extends the concept of conventional convergence and is related to many useful constructions in series summability and probability theory.

It seems natural to try to extend the concept of continuity using statistical convergence instead of conventional convergence. However, this brings us back to the same concept of function continuity as the following explanation demonstrates.

Assume that a function f: R → R is continuous at a point a ∈ R in the conventional sense. Then, it is well-known that f(x) is defined at a and for any sequence l = { a_i ∈ R; i = 1, 2, 3, …} that converges to a, the point f(a) is the limit of the sequence{ f(a_i) ∈ R; i = 1, 2, 3, …} or formally, for any sequence l = { a_i ∈ R; i = 1, 2, 3, …}, the condition lim_i→∞a_i = a implies lim_i→∞f(a_i) = f(a). Using the statistical continuity instead of conventional continuity, we come to the following definition of statistical continuity. A function f: R → R is statistically continuous at a point a ∈ R if f(x) is defined at a and for any sequence l = { a_i ∈ R; i = 1, 2, 3, …} that converges to a, the point f(a) is the statistical limit of the sequence{ f(a_i) ∈ R; i = 1, 2, 3, …}. Recall that a sequence l = {a_i ; i = 1, 2, 3,…} is statistically convergent to a or a = stat-lim l if d({i ∈ N; |a_i - a| ≥ ε }) = 0 for every ε > 0 where the asymptotic density d(K) of a set K ⊆ N is equal to lim_n→∞ (1/n) |K_n| and K_n = {k ∈ K; k ≤ n} (Steinhaus, 1951; Fast, 1951; Schoenberg, 1959; Fridy, 1985). Thus, for a statistically continuous at a point a function f and any sequence l = { a_i ∈ R; i = 1, 2, 3, …} that converges to a, we have that either the sequence{ f(a_i) ∈ R; i = 1, 2, 3, …} converges to f(a) or there are infinitely many elements a_i in l such that | f(a_i) - f(a) | ≥ ε for some ε. If the function f is statistically continuous at the point a but not continuous at this point, then there is a sequence l such that there are infinitely many elements a_ij in l such that | f(a_ij) - f(a) | ≥ ε. Let us take all these elements a_ij. They form a sequence h = { a_ij ∈ R; j = 1, 2, 3, …} and | f(a_ij) - f(a) | ≥ ε for all a_ij, j = 1, 2, 3, … . By the definition of the asymptotic density d(K), we have that d({i ∈ N; |a_ij - a| ≥ ε }) = 1. However, as lim l = a, it has to be d({i ∈ N; |a_ij - a| ≥ ε }) = 0. This contradiction shows that any sequence{ f(a_i) ∈ R; i = 1, 2, 3, …} converges to f(a) when the sequence l = { a_i ∈ R; i = 1, 2, 3, …} that converges to a. Consequently, f is a continuous function in the conventional sense, i.e., the new approach does extend the conventional definition of function continuity.