@article{Hayano_2022, title={Stability of non-proper functions}, volume={128}, url={https://www.mscand.dk/article/view/132211}, DOI={10.7146/math.scand.a-132211}, abstractNote={<p>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.</p>}, number={2}, journal={MATHEMATICA SCANDINAVICA}, author={Hayano, Kenta}, year={2022}, month={Jun.} }