On Morse Theory for Piecewise Smooth Functions

Lower level sets of continuous selections of C2-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 homotopy 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.