arxiv: 2604.26699 · v1 · submitted 2026-04-29 · 🧮 math.CA

Recognition: unknown

Non-symmetrically t-affine functions revisited

D\'ora Koroknai, Tibor Kiss

Pith reviewed 2026-05-07 12:34 UTC · model grok-4.3

classification 🧮 math.CA

keywords non-symmetric t-affine functionsconditional functional equationst-affinityreal subintervalsaffine functionsfunctional equations on intervals

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The pith

A function satisfying the non-symmetric t-affinity condition on any subinterval of the reals must satisfy the full t-affinity equation without the ordering restriction.

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

The paper examines real-valued functions on subintervals of R that obey the conditional equation f(tx + (1-t)y) = t f(x) + (1-t) f(y) only when x ≤ y. It first proves these functions are locally t-affine, so the equality holds for all sufficiently close pairs. From local affinity it derives that the equality holds on entire open intervals. The central result extends the 2014 theorem of Lewicki and Olbryś by showing the implication remains true when the domain is restricted to an arbitrary subinterval rather than the whole real line.

Core claim

If a real-valued function f defined on a subinterval I of R satisfies f(tx + (1-t)y) = t f(x) + (1-t) f(y) for all x, y in I with x ≤ y, then the same equality holds for every pair x, y in I.

What carries the argument

The conditional functional equation restricted to ordered pairs x ≤ y, which is shown to force local t-affinity and thence global t-affinity on any subinterval.

If this is right

The ordering restriction in the hypothesis can be dropped without loss of generality on any subinterval.
Local t-affinity serves as an intermediate step that upgrades to global t-affinity on open intervals.
The earlier result for the whole real line is recovered as the special case when the subinterval is R itself.
No continuity or other regularity assumption is required beyond the conditional equation.

Where Pith is reading between the lines

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

Verification of t-affinity on an interval can be reduced to checking only ordered pairs.
Similar conditional-to-unconditional upgrades might apply to other weighted functional equations.
The result suggests that ordering constraints in functional equations are often removable once local behavior is controlled.

Load-bearing premise

The function is real-valued and satisfies the conditional weighted-average equation for every ordered pair x ≤ y inside its subinterval domain.

What would settle it

Exhibiting a single function on some subinterval of R that meets the conditional equation for all x ≤ y yet fails it for at least one pair with x > y would refute the claim.

read the original abstract

In 2014, Michal Lewicki and Andrzej Olbry\'s proved that if a real valued function $f$ defined on the real line satisfies the conditional functional equation \[ f(tx + (1-t)y) = t f(x) + (1-t) f(y),\qquad x\leq y, \] called non-symmetrically $t$-affine, then it is $t$-affine. That is, they concluded that $f$ must fulfill the above equality without any restriction on $x$ and $y$. In the current study, first we show that the above conditional equation implies that the function in question is locally $t$-affine. Then we derive $t$-affinity on open intervals. Finally, we formulate our main result, which generalizes the theorem of Lewicki and Olbry\'s for any subinterval of $\mathbb{R}$.

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.

Desk Editor's Note private letter to a colleague

This paper extends the 2014 Lewicki-Olbryś theorem on non-symmetrically t-affine functions to arbitrary subintervals via a local-to-open-to-general chain, but the boundary step needs verification.

read the letter

The new piece is the generalization from the full real line to any subinterval of R. They proceed in clear steps: the conditional equation first forces local t-affinity, then t-affinity on open intervals, and finally the main claim that it holds without the x ≤ y restriction on any subinterval. That sequence is a sensible way to build the result and stays close to the original 2014 argument without adding continuity or measurability assumptions, which keeps the statement tight. The citation pattern is straightforward and points directly to the prior work. No free parameters or invented entities show up in the outline. The soft spot is the final step to closed or half-open intervals. The local and open-interval arguments rely on interior points and chaining, but when one or both of x and y sit on the boundary the convex combination tx + (1-t)y can land in ways that the open case does not automatically cover. Without regularity to push density or continuity to the endpoints, that transition is load-bearing and should be checked line by line in the proof. If it is handled explicitly, the claim stands; if it is glossed over, the result is weaker than stated. This is narrow functional-equation work aimed at specialists who already know the 2014 theorem. A reader in that subfield will get a modest but usable extension. It is coherent on its own terms and shows clear thinking, so it deserves a serious referee to confirm the boundary cases rather than a desk rejection.

Referee Report

1 major / 1 minor

Summary. The manuscript generalizes the 2014 theorem of Lewicki and Olbryś by proving that a real-valued function f defined on an arbitrary subinterval I of R and satisfying the conditional equation f(tx + (1-t)y) = t f(x) + (1-t) f(y) for all x ≤ y in I must in fact be t-affine on I (i.e., the equation holds for all x, y in I without the ordering restriction). The argument proceeds in three stages: first establishing local t-affinity, then t-affinity on open intervals, and finally the main result for general subintervals.

Significance. If the boundary-handling details are completed, the result would be a modest but useful extension of the theory of conditional functional equations to restricted domains. The stepwise strategy (local → open intervals → general subintervals) is a standard and potentially reusable technique in real analysis.

major comments (1)

[Main result (as described in the abstract and the final section)] The transition from t-affinity on open intervals to arbitrary subintervals (including closed or half-open ones) is load-bearing for the central claim. The abstract outlines the sequence but does not indicate how the chaining or local arguments are adapted when one or both of x, y lie on the boundary; the convex combination tx + (1-t)y may then lie in a region not covered by the open-interval case, and no regularity assumption (e.g., continuity) is stated to justify density or limit arguments at endpoints.

minor comments (1)

[Abstract] The abstract could explicitly list the types of subintervals (open, closed, half-open) covered by the main theorem to avoid ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The observation regarding the boundary handling in the transition to general subintervals is well taken, and we will revise the manuscript accordingly to strengthen the exposition of the main result.

read point-by-point responses

Referee: [Main result (as described in the abstract and the final section)] The transition from t-affinity on open intervals to arbitrary subintervals (including closed or half-open ones) is load-bearing for the central claim. The abstract outlines the sequence but does not indicate how the chaining or local arguments are adapted when one or both of x, y lie on the boundary; the convex combination tx + (1-t)y may then lie in a region not covered by the open-interval case, and no regularity assumption (e.g., continuity) is stated to justify density or limit arguments at endpoints.

Authors: We agree that the current exposition of the main theorem would benefit from a more explicit treatment of boundary cases. The three-stage argument (local t-affinity, then open intervals, then general subintervals) is intended to proceed without continuity by direct case analysis on the position of x and y relative to the endpoints of I and by repeated application of the conditional equation to suitably ordered interior points. However, we acknowledge that the manuscript does not spell out this adaptation in sufficient detail. In the revised version we will insert a dedicated paragraph in the proof of the main result that enumerates the possible configurations (both endpoints interior, one or both on the boundary) and shows that tx + (1-t)y always falls into a region where t-affinity has already been established, without invoking limits or density. revision: yes

Circularity Check

0 steps flagged

Direct proof chain generalizing prior result by other authors

full rationale

The paper derives the main result via three explicit steps: the given conditional equation implies local t-affinity, which implies t-affinity on open intervals, which extends to arbitrary subintervals of R. This generalizes the 2014 theorem of Lewicki and Olbryś (distinct authors) without any self-citation, fitted parameters, ansatzes, or renaming of known results. No step reduces an output to its input by construction; the argument consists of standard mathematical implications on functional equations and is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard axioms of the real numbers and basic properties of functional equations; no free parameters, invented entities, or ad-hoc assumptions are indicated in the abstract.

axioms (1)

standard math The real numbers form an ordered field with the usual order and arithmetic operations.
Invoked implicitly when discussing intervals and the conditional equation on R.

pith-pipeline@v0.9.0 · 5448 in / 1004 out tokens · 32729 ms · 2026-05-07T12:34:39.453745+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references

[1]

Daróczy, Z., Páles, Zs.,Convexity with given infinite weight sequences, Stochastica 11 (1987), no. 1, 5–12

1987
[2]

Kuczma, M.,An introduction to the theory of functional equations and inequalities, Prace Nauk. Uniw. Śląsk. Katowicach 489 (1985), 1–523

1985
[3]

In- ternat

Kuhn, N.,A note on t-convex functions, General inequalities, 4 (Oberwolfach, 1983), 269–276. In- ternat. Schriftenreihe Numer. Math., 71[International Series of Numerical Mathematics] Birkhäuser Verlag, Basel, 1984

1983
[4]

Lewicki, M., Olbryś, A.,On non-symmetric t-convex functionsMath. Inequal. Appl. 17 (2014), no. 1, p. 95–100. 6

2014
[5]

Exchange 29 (2003/04), no

Nikodem, K., Páles, Zs.,Ont-convex functions, Real Anal. Exchange 29 (2003/04), no. 1, p. 219–228. Institute of Mathematics, University of Debrecen, 4002 Debrecen, Pf. 400, Hungary Email address:kiss.tibor@science.unideb.hu Email address:koroknai.dora1@mailbox.unideb.hu

2003