Morse - Sard Theorem

Let Z andX be Banach spaces, U ⊂ Z be an open convex set and f : U → X be a mapping. We say that f is a d.c. mapping if there exists a continuous convex function h on U such that y ◦ f + h is a continuous convex function for each y ∈ Y , ‖y∗‖ = 1. We say that f : U → X is locally d.c. if for each x ∈ U there exists open convex U ′ such that x ∈ U ′ ⊂ U and f |U ′ is d.c. This notion of d.c. mappings between Banach spaces (see [7]) generalizes Hartman’s [3] notion of d.c. mapping between Euclidean spaces. Note that in this case it is easy to see that F is d.c. if and only if all its components are d.c. (i.e. are differences of two convex functions). For f : U → X we denote Cf := {x ∈ U : f (x) = 0}. A special case of [2, Theorem 3.4.3] says that for a mapping f : R → X of class C, where X is a normed vector space, the set f(Cf) has zero (m/2)dimensional Hausdorff measure. We will generalize this result in case m = 1 showing that it is sufficient to suppose that f is d.c. on I (equivalently: f is continuous and f ′ + is locally of bounded variation on I). Similar generalization of above mentioned result on C mappings holds for m = 2 as is shown (by completely different method) in [6] where it is proved that f(Cf) has zero 1-dimensional Hausdorff measure for d.c. mapping f :