{"paper":{"title":"Nonsmooth Morse-Sard theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.OC"],"primary_cat":"math.CA","authors_text":"Daniel Azagra, Javier Gomez-Gil, Juan Ferrera","submitted_at":"2016-05-05T07:29:57Z","abstract_excerpt":"We prove that every function $f:\\mathbb{R}^n\\to \\mathbb{R}$ satisfies that the image of the set of critical points at which the function $f$ has Taylor expansions of order $n-1$ and non-empty subdifferentials of order $n$ is a Lebesgue-null set. As a by-product of our proof, for the proximal subdifferential $\\partial_{P}$, we see that for every lower semicontinuous function $f:\\mathbb{R}^2\\to\\mathbb{R}$ the set $f(\\{x\\in\\mathbb{R}^2 : 0\\in\\partial_{P}f(x)\\})$ is $\\mathcal{L}^{1}$-null."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.01513","kind":"arxiv","version":2},"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"}