 
Summary: On Morse Theory for Piecewise Smooth Functions
Andrei A. Agrachev, Diethard Pallaschke, and Stefan Scholtes
Abstract
Lower level sets of continuous selections of C 2 functions dened 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.
Key Words: Continuous selection, critical point, nonsmooth Morse theory
AMS(MOS) Subject Classication: 26A27, 58C27, 90C30
1 Introduction
For a smooth function f : M ! IR dened on a smooth ndimensional manifold M;
Morse Theory studies the topological types of the lower level sets
M = f x j f(x) g
for 2 IR (cf. [13]). The rst Morse Lemma states that for a C 1 function f and <
the lower level set M is a strong deformation retract of M provided that M is compact
and f 1 ([; ]) does not contain a critical point. The second Morse Lemma is concerned
with the local behaviour of C 2 functions in a neighborhood of a nondegenerated critical
point x 0 2 M . It states that whenever the Hessian of f at x 0 is regular, the function f is
