Abstract

The goal of this work is to introduce and study fuzzy limits of functions. Two approaches to fuzzy limits of a function are considered. One is based on the concept of a fuzzy limit of a sequence, while another generalizes the conventional ε-δ definition. It is demonstrated that these constructions are equivalent. Different properties of fuzzy limits of functions are found. Properties of conventional limits are easily deduced from properties of fuzzy limits. In the second part of this work, the concept of fuzzy limits of a function is extended to provide means to define nontrivial continuity of functions on discrete sets. In addition, fuzzy limits of functions are introduced and studied.

Key words:

fuzzy limit, fuzzy convergence, sequence, function

1. Introduction

Mathematics is an efficient tool for modeling real world phenomena. However, in its essence mathematics is opposite to real world because mathematics is exact, rigorous and abstract while real things and systems are imprecise, vague, and concrete. To lessen this gap, mathematicians elaborate methods that make possible to work with natural vagueness and incompleteness of information using exact mathematical structures. One of the most popular approaches to this problem is fuzzy set theory.

2. Fuzzy limits of sequences

The theory of fuzzy limits of functions is based on the theory of fuzzy limits of sequences. That is why we begin our exposition with the main concepts and constructions from the latter theory.

Let r∈R+ and l = {a i ∈ R; i∈ω} be a sequence of real numbers.

Definition 2.1. A number a is called an r-limit of a sequence l (it is denoted by a = r-lim i→∞ ai or a = r-lim l) if for any ε∈ R++ the inequality ρ(a, a i) < r + ε is valid for almost all a i, i.e., there is such n that for any i > n, we have ρ(a, a i) < r + ε.

In this case, we say that l r-converges to a and denote it by l →q a.

By extending what we know, we also have, Definition 2.2. a) A number a is called a fuzzy limit of a sequence l if it is an r-limit of l for some r∈R+. b) a sequence l fuzzy converges if it has a fuzzy limit.

This theory and analysis provide more flexible means for the development of the differential calculus and differential equations for functions defined in discrete spaces than traditional approach based on finite differences.

Denotations

The understanding and application of these concept necessitates familiarizing oneself with a set of symbols, formulas and terms. For better understanding, below are the denotations used in the context.

• N is the set of all natural numbers;
• ω is the sequence of all natural numbers;
• Z is the set of all integer numbers;
• ∅ is the empty set;
• R is the set of all real numbers;
• R+ is the set of all non-negative real numbers;
• R++ is the set of all positive real numbers;
• R∞ = R ∪{∞, - ∞ };
• if a is a real number, then |a| or ||a|| denotes its absolute value or modulus;
• if a is a real number and t ∈R++, then Ota = { x ∈R; a – t < x < a + t } is a neighborhood of a;
• ρ(x,y) = | x - y | for x,y∈ R;

These serve as the base to understand, and further elaborate the concept of fuzzy limits of functions and sequences. The development of the differential calculus and differential equations for functions defined in discrete spaces are thus better understood and defined with the concept of fuzzy limits.