{"paper":{"title":"Topological stability of continuous functions with respect to averaging by measures with locally constant densities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.GN"],"primary_cat":"math.CA","authors_text":"Oksana Marunkevych, Sergiy Maksymenko","submitted_at":"2016-01-02T07:36:24Z","abstract_excerpt":"Let $\\mu$ be a measure on $[-1,1]$. Then for every continuous function $f:\\mathbb{R}\\to\\mathbb{R}$ and $\\alpha>0$ one can define its averaging $f_{\\alpha}:\\mathbb{R}\\to\\mathbb{R}$ by the formula: \\[ f_{\\alpha}(x) = \\int_{-1}^{1} f(x+t\\alpha)d\\mu. \\] In arXiv:1509.06064 the authors studied the problem when $f_{\\alpha}$ is topologically equivalent to $f$ for all $\\alpha>0$ and call this property a topological stability of $f$ under averagings with respect to measure $\\mu$. Similarly one can define topological stability of a germ of $f$ at some point $x\\in\\mathbb{R}$.\n  It was shown that for a co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00151","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"}