{"paper":{"title":"Some remarks about The Morse-Sard theorem and approximate differentiability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Daniel Azagra, Miguel Garc\\'ia-Bravo","submitted_at":"2017-05-16T10:03:27Z","abstract_excerpt":"Let $n, m$ be positive integers, $n\\geq m$. We make several remarks on the relationship between approximate differentiability of higher order and Morse-Sard properties. For instance, among other things we show that if a function $f:\\mathbb{R}^n\\to\\mathbb{R}^m$ is locally Lipschitz and is approximately differentiable of order $i$ almost everywhere with respect to the Hausdorff measure $\\mathcal{H}^{i+m-2}$, for every $i=2, \\dots, n-m+1$, then $f$ has the Morse-Sard property (that is to say, the image of the critical set of $f$ is null with respect to the Lebesgue measure in $\\mathbb{R}^m$)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.05624","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}