Lower level sets of continuous selections of functions defined on a smooth manifold in the vicinity of a nondegenerate critical point in the sense of  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.
Title: On Morse theory for piecewise smooth functions
Authors: A.A. Agrachev, S. Scholtes, D. Pallaschke
Journal title: Dynamical and Control Systems