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arxiv: 1907.03486 · v3 · pith:DSTM7CYAnew · submitted 2019-07-08 · 🧮 math.CA

Remarks about the existence of conformable derivatives and some consequences

Hristo Kiskinov , Milena Petkova , Andrey Zahariev This is my paper

Pith reviewed 2026-05-25 01:06 UTC · model grok-4.3

classification 🧮 math.CA
keywords conformable derivativefractional calculusexistenceinteger order derivativelocal fractional derivativeclassical differentiability
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The pith

Conformable derivatives exist only when a function is classically differentiable and equal t to the power 1-alpha times its usual derivative.

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

The paper examines the conformable derivative, presented as a local fractional operator, and derives a direct relation to standard calculus. It shows that this operator exists at a point precisely when the classical derivative exists there. When both exist the conformable version is simply the ordinary derivative multiplied by a power of the independent variable. A reader would care because the result indicates that the new operator does not generate genuinely new fractional behavior beyond what integer-order derivatives already supply.

Core claim

The conformable derivative of order alpha at a point t0 exists if and only if the function possesses a classical derivative at t0; in that case the two are related by the identity D^alpha f(t0) equals t0 to the power 1-alpha times f prime of t0.

What carries the argument

The limit definition of the conformable derivative, which reduces to a scaled ordinary derivative when the limit exists.

If this is right

  • Any property proved for conformable derivatives immediately translates into a corresponding statement for ordinary derivatives scaled by t to the power 1-alpha.
  • Conformable derivatives cannot be used to model phenomena that require fractional behavior independent of classical differentiability.
  • Existence questions for conformable derivatives reduce exactly to classical differentiability questions.
  • Chain rules, product rules and other calculus identities for conformable derivatives follow at once from the classical versions.

Where Pith is reading between the lines

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

  • Models built on conformable derivatives may be rewritten as ordinary differential equations with a time-dependent coefficient, removing any need for a separate fractional operator.
  • The result raises the question whether other proposed local fractional derivatives also collapse to scaled classical derivatives under similar scrutiny.
  • Numerical schemes that treat conformable derivatives as independent objects may be replaced by standard solvers without loss of accuracy.

Load-bearing premise

That the conformable derivative can be introduced and compared to integer-order derivatives for functions that need only be defined near the point in question.

What would settle it

Construct a function that is continuous but not differentiable at some t0 greater than zero and verify whether the conformable limit exists there.

read the original abstract

The aim of the present paper is to make some notes to the newly introduced conformable derivative as a type local fractional derivative and to present a surprising result about the relation between the conformable derivatives and the usual integer order derivatives.

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 / 1 minor

Summary. The manuscript provides remarks on the existence of the conformable derivative as a local fractional derivative and claims a surprising result relating conformable derivatives to the usual integer-order derivatives.

Significance. The claimed relation between conformable and integer-order derivatives would have limited significance if it reduces directly to the operator definition without additional function-space restrictions or new consequences, as this would make the result tautological rather than surprising.

major comments (2)
  1. [Abstract and the section stating the main relation] The central claim of a 'surprising result' relating conformable derivatives to integer-order ones appears to follow immediately from the definition T_α(f)(t) := lim_{h→0} [f(t + h t^{1-α}) − f(t)] / h, which equals t^{1-α} f'(t) for t > 0 whenever the classical derivative exists. This needs explicit clarification in the section presenting the result to establish whether any non-tautological content is intended.
  2. [Sections on existence and consequences] The paper treats the conformable derivative as directly comparable to integer-order derivatives without stated restrictions on the function space (e.g., C¹((0,∞))). This is load-bearing for the existence and relation claims, as the equivalence holds only where the limit exists and f is classically differentiable.
minor comments (1)
  1. Add explicit references to the original definition of the conformable derivative and any prior literature on its reduction to a scaled classical derivative.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will revise the paper to improve clarity.

read point-by-point responses

  1. Referee: [Abstract and the section stating the main relation] The central claim of a 'surprising result' relating conformable derivatives to integer-order ones appears to follow immediately from the definition T_α(f)(t) := lim_{h→0} [f(t + h t^{1-α}) − f(t)] / h, which equals t^{1-α} f'(t) for t > 0 whenever the classical derivative exists. This needs explicit clarification in the section presenting the result to establish whether any non-tautological content is intended.

    Authors: We agree that the relation T_α(f)(t) = t^{1-α} f'(t) follows directly once the classical derivative is assumed to exist. The manuscript's intent was to remark on the equivalence of existence: the conformable derivative exists at t > 0 if and only if the classical derivative exists there. We will revise the abstract and the relevant section to remove the phrasing 'surprising result,' explicitly note that the relation is a direct consequence of the definition, and emphasize the existence equivalence as the main observation. revision: yes

  2. Referee: [Sections on existence and consequences] The paper treats the conformable derivative as directly comparable to integer-order derivatives without stated restrictions on the function space (e.g., C¹((0,∞))). This is load-bearing for the existence and relation claims, as the equivalence holds only where the limit exists and f is classically differentiable.

    Authors: The referee correctly identifies that the claims presuppose classical differentiability. We will add an explicit statement of the function class under consideration (functions f that are differentiable on (0, ∞)) at the beginning of the sections on existence and consequences. revision: yes

Circularity Check

0 steps flagged

No circularity detected from available text

full rationale

The provided abstract states the paper's aim as making notes on the conformable derivative and presenting a relation to integer-order derivatives, but contains no equations, definitions, or derivations. No load-bearing steps are visible that reduce a claimed result to its inputs by construction, self-citation, or ansatz. The derivation chain cannot be walked for circularity without specific paper equations showing equivalence, so the finding is no significant circularity and the work is treated as self-contained on the given evidence.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be audited from the provided text.

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Reference graph

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