Arc-Smooth and Whitney-Smooth Functions on Closed Sets

Definitions and mathematical statement

The supplied problem asks for conditions ensuring the equalities \[ A^\infty(X)=C^\infty(X),\qquad A^M(X)=C^M(X), \] for non-open subsets $X\subseteq{\mathbb{R}}^d$. These are characterization questions, not assertions that the equalities hold for every such set. The source does not fully specify the four classes, so the following definitions explain a possible scalar-valued interpretation and identify the conventions still requiring confirmation.

A smooth arc in $X$ is a map $c:I\to{\mathbb{R}}^d$, where $I\subseteq{\mathbb{R}}$ is an open interval, such that $c$ is smooth and $c(I)\subseteq X$. For real-valued functions, the usual smooth-arc testing class is \[ \mathcal A^\infty(X)=\{f:X\to{\mathbb{R}}: f\circ c\in C^\infty(I)\text{ for every smooth arc }c:I\to X\}. \] One possible interpretation of smoothness on $X$ is extension to an open neighborhood: \[ \mathcal E^\infty(X)=\{f:X\to{\mathbb{R}}:\text{ there exist an open }U\supseteq X\text{ and }F\in C^\infty(U)\text{ with }F|_X=f\}. \] If the intended conventions are $A^\infty(X)=\mathcal A^\infty(X)$ and $C^\infty(X)=\mathcal E^\infty(X)$, the smooth question is to characterize those non-open $X$ for which every function passing all smooth-arc tests admits such an extension. This interpretation is conditional: the supplied record does not specify the codomain or whether its Whitney-type class means extendible functions, functions admitting compatible jets, or jets themselves. A Whitney jet consists of functions $f_\alpha:X\to{\mathbb{R}}$, indexed by derivative multi-indices $\alpha$, subject to the appropriate Taylor remainder compatibility conditions; it contains more data than its zeroth component $f_0$. Thus a comparison with a class of functions requires an explicit convention about passing from jets to their zeroth components.

For the ultradifferentiable question, $M$ denotes an unspecified ultradifferentiable datum. Once an ambient class $C^M$, a family $\mathscr T^M(X)$ of testing arcs with their open parameter intervals, and a Whitney-type class on $X$ have been fixed, the arc-testing construction has the form \[ \mathcal A^M_{\mathscr T}(X)=\{f:X\to{\mathbb{R}}: f\circ c\in C^M(I)\text{ for every }(I,c)\in\mathscr T^M(X)\}. \] The intended task is to determine hypotheses on $X$ and $M$ under which this testing class agrees with the intended $C^M(X)$. The record does not determine whether the testing arcs are smooth or ultradifferentiable, the growth estimates defining $C^M$, the assumptions on $M$, or the Roumieu versus Beurling convention. Consequently it does not determine a single precise ultradifferentiable equality without further source clarification.

The title refers to closed sets, whereas the statement explicitly allows non-open subsets without requiring closedness. Closedness must therefore remain an unresolved scope qualification, rather than an added hypothesis.

Short English statement

Determine when testing a function along arcs detects exactly the intended Whitney-type smoothness on a non-open subset of Euclidean space, both in the smooth and ultradifferentiable settings. The meanings of the Whitney-type classes, the ultradifferentiable datum $M$, and the permitted testing arcs require clarification. The supplied record labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003499$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Title: Arc-Smooth and Whitney-Smooth Functions on Closed Sets. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied source citation: O-Minimality and its Applications to Number Theory and Analysis (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/22, pp. 1349--1420. https://doi.org/10.4171/owr/2017/22. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The literature-source label is "Oberwolfach Reports 2018." The supplied 2018 designation and the 2017 proposed year and DOI component are retained without resolving their chronology.
• Supplied literature reference: Rainer, arc-smooth functions on closed sets. https://arxiv.org/abs/1801.08335.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports results for important non-open domains, including closed fat sets and subanalytic settings, while describing arbitrary non-open subsets and the full ultradifferentiable framework as unresolved. This is supplied dated curation derived from the OWR record, not an independently verified assessment. Time-sensitive status and the cited mathematical claims require independent verification.