Research Problems in Function Theory — Problem 7.77

Definitions and mathematical statement

Let $I\subset{\mathbb{R}}$ be a finite interval of positive length and $m$ Lebesgue measure. Real measurable functions $f,g$ on $I$ are rearrangements of each other when \[ m\{x\in I:g(x)<y\}=m\{x\in I:f(x)<y\} \qquad\text{for every }y\in{\mathbb{R}}. \] Equivalently, they have equal pushforward measures on ${\mathbb{R}}$ under $m|_I$. For a class of real functions $f$ with a measurable derivative $f'$ defined at least almost everywhere, Rubel asks to classify those satisfying \[ m\{x\in I:f'(x)<y\}=m\{x\in I:f(x)<y\} \qquad(y\in{\mathbb{R}}). \] Values of $f'$ at the interval endpoints or other null exceptional sets do not affect this condition.

The source examples are $f(x)=ke^x$ on any finite interval and $f(x)=k\sin x$ on $[0,\pi/2]$, with real constant $k$. It asks what additional functions occur, and more broadly asks about other "differential rearrangements." That additional program does not name a differential operator or relation; no specific higher-derivative or differential-equation target is invented.

The excerpt does not specify whether the classification is among $C^1$, everywhere differentiable, absolutely continuous, or merely almost-everywhere differentiable functions, nor whether the interval is fixed or may vary as part of the classification. These give materially different classes, so the displayed equidistribution property is fixed while the missing regularity scope remains explicit. The final printed $f'(z)\sim f(x)$ is a variable mismatch in a real-interval question, not an instruction to add a complex-analytic domain. The supplied OPEN-TRIAGE summary reports open status in the 2018 edition and no definitive resolution located.

Short English statement

Classify real functions on a finite interval whose derivative has exactly the same value distribution as the function itself. The derivative regularity and the intended broader differential-rearrangement questions are unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $2307077$, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 7.77."
• Hayman, Research Problems in Function Theory (2018), Problem 7.77, attributed to L. A. Rubel; https://arxiv.org/abs/1809.07200. Source-TeX extraction; recorded access 2026-07-29.
• Supplied OPEN-TRIAGE summary, without a separate authoritative current-resolution citation.