Some new perspectives on Sard's theorem

Let E be a vector space of finite dimension, F a normed vector space, k gre ater than or equal to 1 an integer, f : E --> F a C-k function, t > 0 a rea l number and A subset of E a bounded subset with finite H-t-measure, where H-t is the Hausdorff measure of dimension t. If the differential of f in ev ery point of A has rank less than or equal to r, we show that the image f(A ) has zero H-d-measure, where d = r + t-r/k. Moreover if D(k)f satisfies a Holder condition of exponent alpha is an element of (0, 1]; then f(A) has f inite H-d-measure, where d = r + t-r/k+alpha. If E is a Banach space, then the preceding results are still true if k less than or equal to 2. On the other side, we present a polynomial function P : l(2) --> R for which the critical set has finite H-3-measure, and whose i mage is [0, 1]. (C) 2001 Academie des sciences/Editions scientifiques et me dicales Elsevier SAS.