Stability of non-proper functions
DOI:
https://doi.org/10.7146/math.scand.a-132211Abstract
The purpose of this paper is to give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney topology). We show that a Morse function is stable if it is end-trivial at any point in its discriminant, where end-triviality (which is also called local triviality at infinity) is a property concerning behavior of functions around the ends of the source manifolds. We further show that a Morse function is strongly stable if (and only if) it is quasi-proper. This result yields existence of a strongly stable but not infinitesimally stable function. Applying our result on stability, we give a sufficient condition for stability of Nash functions, and show that any Nash function becomes stable after a generic linear perturbation.
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