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The purpose of this paper is to give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney C–topology). We introduce the notion of end-triviality of smooth mappings, which concerns behavior of mappings around the ends of the source manifolds, and show that a Morse function is stable if it is end-trivial at any point in its discriminant. We further show that a Morse function f: N → R is strongly stable (i.e. there exists a continuous mapping g 7→ (Φg, φg) ∈ Diff(N)×Diff(R) such that φg ◦g ◦Φg = f for any g close to f) if (and only if) f is quasi-proper. This result yields existence of a strongly stable but not infinitesimally stable function. Applying our result on stability, we also show that a locally stable Nash function on Rn is stable if it satisfies some mild condition on its gradient, and as a corollary, that the following non-proper function is stable (where k ∈ {1, . . ., n − 1}):

57R45, 14P20, 58D99

Original languageEnglish
JournalUnknown Journal
Publication statusPublished - 2018 Sep 7


  • Stability of smooth mappings

ASJC Scopus subject areas

  • General

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