[Paper Review] On the space of Type-2 interval with limit, continuity and differentiability of Type-2 interval-valued functions

Sadikur Rahman, Ali Akbar Shaikh+1 more|arXiv (Cornell University)|Jul 1, 2019

Fuzzy Systems and Optimization2 references4 citations

TL;DR

This paper introduces Type-2 interval-valued functions, where both endpoints are intervals, and establishes a complete metric space using an extended Moore distance. It defines limit, continuity, and gH-differentiability for these functions, proving the space is complete and deriving foundational properties for generalized Hukuhara differentiability in Type-2 settings.

ABSTRACT

This paper deals with the new concept of interval whose both the bounds themselves are also intervals. We name this new type of interval as Type-2 interval. Here we have introduced Type-2 interval-valued function and its properties. To serve this purpose, we have defined a distance on the set of all Type-2 intervals, named as extended Moore distance for Type-2 intervals which is a metric on the set of all Type-2 intervals. Then we have shown that the space of Type-2 interval is a complete metric space with respect to extended Moore distance. Then we have introduced the concept of limit-continuity for Type-2 interval-valued function of single variable and also, we have derived some elementary properties of this concept. Subsequently, we have presented the idea of generalised Hukuhara difference on the set of Type-2 intervals. Finally, using this difference, we have defined gH-differentiability of Type-2 interval-valued function and discussed some of its properties.

Motivation & Objective

• To formalize a new class of intervals—Type-2 intervals—where both bounds are intervals, enabling modeling of higher-order uncertainty.
• To define a metric on the set of Type-2 intervals, named the extended Moore distance, to support topological and analytical operations.
• To establish the space of Type-2 intervals as a complete metric space under the extended Moore distance.
• To introduce the concepts of limit and continuity for Type-2 interval-valued functions of a single variable.
• To define generalized Hukuhara differentiability for Type-2 interval-valued functions and explore its properties.

Proposed method

• The extended Moore distance is defined as a metric on the set of all Type-2 intervals, ensuring the space is complete.
• The paper constructs the space of Type-2 intervals and proves it is complete under the extended Moore distance.
• Limit and continuity for Type-2 interval-valued functions are defined analogously to real-valued functions but adapted to interval-valued bounds.
• Generalized Hukuhara difference is introduced on the set of Type-2 intervals to support derivative definitions.
• gH-differentiability is defined using the generalized Hukuhara difference, extending classical differentiability to Type-2 interval-valued functions.
• Elementary properties of gH-differentiability are derived, including linearity and chain rule analogs under specific conditions.

Experimental results

Research questions

• RQ1 Can a complete metric space be constructed for Type-2 intervals, where both endpoints are intervals?
• RQ2 How can the concepts of limit and continuity be meaningfully extended to Type-2 interval-valued functions?
• RQ3 What is the appropriate generalization of the Hukuhara difference for Type-2 intervals to enable differentiability?
• RQ4 How can gH-differentiability be defined and what properties does it satisfy in the context of Type-2 interval-valued functions?
• RQ5 What is the role of the extended Moore distance in ensuring topological and analytical consistency in Type-2 interval spaces?

Key findings

• The space of Type-2 intervals equipped with the extended Moore distance forms a complete metric space.
• The extended Moore distance satisfies all metric space axioms, enabling rigorous analysis of convergence and continuity.
• Limit and continuity for Type-2 interval-valued functions are well-defined and satisfy analogs of classical real analysis properties.
• The generalized Hukuhara difference is introduced and shown to be consistent with the structure of Type-2 intervals.
• gH-differentiability is defined for Type-2 interval-valued functions, and its basic properties, such as linearity, are derived.
• The framework provides a foundation for further development of calculus on Type-2 interval-valued functions, including integration and optimization under uncertainty.