Discontinuous Analysis: Peer-Reviewed Publication and Research Status

Discontinuous Analysis is a mathematical framework developed by Victor Porton and presented in a peer-reviewed journal article published in the Journal of Analysis & Number Theory. The framework extends familiar operations of mathematical analysis to generalized settings intended to cover arbitrary, including discontinuous, functions.

Peer-reviewed publication

The principal published reference is:

V. L. Porton, “Discontinuous Analysis,” Journal of Analysis & Number Theory, Vol. 14, No. 1, pp. 1–15, 2026.
DOI: 10.18576/jant/140101

The publisher records the article as received on 7 July 2025, revised on 21 September 2025, accepted on 23 November 2025, and published online on 1 January 2026. The publisher’s article page provides the bibliographic record and abstract.

This publication is important to the status of the project: Discontinuous Analysis is not merely an unpublished proposal described by its author. A substantive presentation of the framework has appeared in a mathematical journal after peer review. At the same time, publication of one paper should not be confused with broad independent validation of every claim, extension, or proposed application of the framework.

What the published work develops

The paper studies a generalized notion of the limit of an arbitrary function using the author’s theory of funcoids. On this basis, it develops or motivates generalized versions of familiar constructions including:

• limits intended to be defined for arbitrary functions at arbitrary points;
• generalized derivatives;
• generalized integrals;
• generalized sums of series;
• algebraic operations on generalized values;
• a notion of generalized or non-classical solution for differential equations.

One of the central ideas is to enlarge the space in which limits and related operators take their values, rather than requiring every generalized value to be an ordinary real number, complex number, or vector. The paper argues that this allows linearity and algebraic identities to be retained in circumstances where a classical limit, derivative, or integral would not exist in the ordinary sense.

Relation to generalized functions and nonsmooth analysis

The published article compares the proposed framework with approaches based on distributions and other forms of nonsmooth analysis. In particular, it argues that its generalized objects permit certain algebraic operations, including multiplication, more broadly than standard distribution theory does.

These are mathematical claims made and developed in the paper. Their scope, advantages, and relationship to established theories should be assessed by examining the definitions and proofs and by comparing them with the existing literature on generalized functions, distributions, nonlinear generalized functions, nonsmooth analysis, and related subjects.

What publication establishes—and what it does not

The journal publication establishes an identifiable scholarly record for the theory: a citable article with a permanent DOI, publication metadata, and a published mathematical exposition. It therefore provides a concrete primary source that researchers can read, cite, criticize, reproduce, or build on.

It does not by itself establish that Discontinuous Analysis has achieved broad acceptance in mathematics, that every proposed extension is correct, or that the framework is superior to all existing approaches. Those stronger conclusions require continued mathematical scrutiny, comparison with prior work, independent use, citations, and further results.

The page also does not rely on a previous Navier–Stokes proof attempt by the author. That proof attempt was found to be incorrect and should not be used as evidence for the validity of Discontinuous Analysis. The relevant evidence for this framework is its mathematical content, including the peer-reviewed article cited above.

How to evaluate the work

Readers interested in the theory should begin with the published paper and evaluate the definitions, proofs, and examples directly. Useful next steps include:

• checking the generalized-limit construction and its algebraic properties;
• comparing the resulting derivatives and integrals with established generalized-function frameworks;
• testing the theory on concrete discontinuous or singular examples;
• looking for counterexamples, hidden assumptions, or equivalences with existing constructions;
• developing additional applications and independent proofs;
• publishing independent reviews, criticism, or extensions.

Science DAO’s broader position is that unconventional or independent mathematical research should be evaluated on its mathematical content and evidence rather than accepted or rejected because of the author’s institutional status.

Citation

Porton, V. L. (2026). “Discontinuous Analysis.” Journal of Analysis & Number Theory, 14(1), 1–15. https://doi.org/10.18576/jant/140101.