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arxiv: 1907.00644 · v1 · pith:3UZ6RIORnew · submitted 2019-07-01 · 🧮 math.FA

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

Md Sadikur Rahman , Ali Akbar Shaikh , Asoke Kumar Bhunia This is my paper

Pith reviewed 2026-05-25 11:43 UTC · model grok-4.3

classification 🧮 math.FA
keywords Type-2 intervalextended Moore distancecomplete metric spacegeneralised Hukuhara differencegH-differentiabilityinterval-valued functionlimitcontinuity
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The pith

The space of Type-2 intervals is a complete metric space under the extended Moore distance, supporting definitions of continuity and gH-differentiability for Type-2 interval-valued functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces a new type of interval called Type-2 interval, in which both the lower and upper bounds are themselves intervals. It constructs an extended Moore distance on these intervals and proves that the collection of all Type-2 intervals forms a complete metric space with respect to this distance. The authors then define the limit and continuity of Type-2 interval-valued functions and derive some of their basic properties. They further introduce the generalised Hukuhara difference for Type-2 intervals and use it to define gH-differentiability, discussing several properties of this derivative.

Core claim

The central claim is that the set of Type-2 intervals with the extended Moore distance is a complete metric space. Using the generalised Hukuhara difference on this set, the paper defines gH-differentiability for Type-2 interval-valued functions of one variable and examines the associated concepts of limit and continuity along with their elementary properties.

What carries the argument

The extended Moore distance on Type-2 intervals, which serves as a metric making the space complete, and the generalised Hukuhara difference, which enables the definition of differentiability.

If this is right

  • The completeness of the space allows for the study of convergent sequences of Type-2 intervals.
  • Limit and continuity concepts provide a foundation for analyzing Type-2 interval-valued functions.
  • gH-differentiability extends the idea of derivatives to these functions using the generalised difference.
  • Properties derived from these definitions hold for single-variable functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could be used to model situations where interval uncertainty itself has uncertainty.
  • The metric space structure might support fixed-point theorems or other analysis tools in this setting.
  • Extensions to multi-variable functions or integration could follow similar lines.
  • Applications in uncertain dynamical systems with nested uncertainties may become possible.

Load-bearing premise

The extended Moore distance is assumed to be a true metric on the set of Type-2 intervals, and the generalised Hukuhara difference is assumed to exist in a way that supports a consistent derivative definition.

What would settle it

Finding two Type-2 intervals whose extended Moore distance violates the triangle inequality, or a pair where no generalised Hukuhara difference exists, would falsify the foundational claims.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces the concept of Type-2 intervals (intervals whose endpoints are themselves intervals). It defines an extended Moore distance on this set, claims the distance is a metric, and asserts that the resulting space is complete. The manuscript then defines notions of limit and continuity for Type-2 interval-valued functions of one variable, introduces a generalised Hukuhara difference on Type-2 intervals, and uses it to define gH-differentiability together with some elementary properties.

Significance. If the metric axioms hold and the generalised Hukuhara difference is consistently defined, the work supplies a complete metric space and a derivative notion for a higher-order interval object. This could support uncertainty modeling with nested intervals. The completeness result and the gH-differentiability definition would constitute the main technical contributions, provided the supporting arguments are supplied and verified.

major comments (2)
  1. [Section introducing the extended Moore distance] The claim that the extended Moore distance is a metric (and in particular satisfies the triangle inequality) is load-bearing for the completeness theorem and all later results on functions; the manuscript must supply an explicit verification rather than an assertion.
  2. [Section on the generalised Hukuhara difference] The definition of the generalised Hukuhara difference for Type-2 intervals must be shown to be well-defined for all admissible pairs and to interact correctly with the metric; without this, the subsequent gH-differentiability notion rests on an unverified axiom.
minor comments (2)
  1. [Throughout] Standardise notation for Type-2 intervals (e.g., consistent use of brackets or subscripts) to improve readability.
  2. [Introduction and definitions] Add citations to the classical literature on Moore metric, gH-differentiability, and interval-valued functions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We address each point below and will revise the manuscript to supply the requested verifications.

read point-by-point responses

  1. Referee: [Section introducing the extended Moore distance] The claim that the extended Moore distance is a metric (and in particular satisfies the triangle inequality) is load-bearing for the completeness theorem and all later results on functions; the manuscript must supply an explicit verification rather than an assertion.

    Authors: We agree that an explicit verification of all metric axioms, especially the triangle inequality, is necessary. In the revised manuscript we will insert a complete, self-contained proof that the extended Moore distance satisfies positivity, symmetry and the triangle inequality. This will directly underpin the completeness theorem and the subsequent results on limits and differentiability. revision: yes

  2. Referee: [Section on the generalised Hukuhara difference] The definition of the generalised Hukuhara difference for Type-2 intervals must be shown to be well-defined for all admissible pairs and to interact correctly with the metric; without this, the subsequent gH-differentiability notion rests on an unverified axiom.

    Authors: We acknowledge the need to prove that the generalised Hukuhara difference is well-defined on all admissible pairs and that it is compatible with the extended Moore metric. The revised version will contain a detailed argument establishing these properties, including explicit conditions under which the difference exists and a verification that the metric is preserved under the operation. This will place the gH-differentiability definition on a rigorous footing. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces new definitions (Type-2 intervals, extended Moore distance, gH-difference) and proves standard metric properties (completeness) plus derived notions (limit, continuity, differentiability) from those definitions. No step reduces a claimed result to a fitted parameter, self-citation chain, or input by construction; all load-bearing claims are explicit constructions or direct consequences of the metric axioms and difference operation as stated. The work is self-contained against external benchmarks and contains no self-definitional loops or renamed known results presented as derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 3 invented entities

The central claims rest entirely on newly introduced definitions whose metric and difference properties are asserted without external benchmarks or independent evidence.

axioms (2)
  • ad hoc to paper The extended Moore distance is a metric on the set of Type-2 intervals.
    Invoked to prove the space is complete.
  • ad hoc to paper The generalised Hukuhara difference exists and is suitable for defining differentiability on Type-2 intervals.
    Required to introduce the derivative concept.
invented entities (3)
  • Type-2 interval no independent evidence
    purpose: To represent nested interval uncertainty
    Newly defined structure with no independent evidence supplied.
  • extended Moore distance for Type-2 intervals no independent evidence
    purpose: To equip the set with a metric
    Newly defined distance with no external validation.
  • generalised Hukuhara difference for Type-2 intervals no independent evidence
    purpose: To enable subtraction and differentiation
    Extension of an existing operation to the new domain without independent checks.

pith-pipeline@v0.9.0 · 5715 in / 1344 out tokens · 62854 ms · 2026-05-25T11:43:52.387005+00:00 · methodology

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