On Transformations in Rn And a Theorem of Sard

Abstract

Extensions of Theorem B under less restrictive conditions have been given by Rado and Reichelderfer, and in particular it has been shown that Theorem B holds if f is merely differentiable on D ([2], pp. 339, 343). Here, however, we restrict ourselves to the case stated above, which is the case most often used in the theory of differentiable manifolds. The result of Theorem B is most naturally viewed as an extension of the inequality of Theorem A to the case in which J(x) vanishes at points of D, and indeed Theorem B shows that (1.1) continues to hold in this case. It is also clear that we can remove the hypothesis in Theorem A that f is (1-1), for if f is not (1-1) the integral fBI J(x) j dx is equal to the measure of f(E) with multiplycovered volumes being counted multiply. We are therefore led to the following theorem, which contains both Theorems A and B.