Title: On Morse theory for piecewise smooth functions
Authors: A.A. Agrachev, S. Scholtes, D. Pallaschke
Journal title: Dynamical and Control Systems
Year: 1997
Volume: 3
Pages: 449-469
Abstract:

Lower level sets of continuous selections of $$C^2$$ functions defined on a smooth manifold in the vicinity of a nondegenerate critical point in the sense of [11] are studied. It is shown that the lower level set is homotropy equivalent to the join of the lower level sets of the smooth and the nonsmooth part, respectively, of the corresponding normal form. Some generalized Morse inequalities are deduced from this result.