THE MORSE-SARD THEOREM WITH MIXED DIFFERENTIABILITY

Suppose that f: M(m) x OMEGA(k) --> N(n) is of differentiablity class C(r x w) in the sense that f = f (x,y) is jointly C(r) in x and analytic in y. If r greater-than-or-equal-to m it is shown that the Morse-Sard conclusion remains true-the set of critical values of f has n-measure zero. In this context the inequality r greater-than-or-equal-to m is sharp; the value of k is irrelevant.