FUNCTIONS OF 2 VARIABLES WITH LARGE TANGENT PLANE SETS

We show that there exist a C-1 function, f, of two variables and a set E subset of or equal to R-2 of zero Lebesgue measure such that using the natural three-dimensional parametrization of planes z = ax + by c tangent to the surface z = f(x, y), the (three-dimensional) interior of the set of parameter values, (a, b, c), of tangent planes correspo nding to points (x, y) in E is nonempty. From the Morse-Sard theorem i t follows that there are no such C-2 functions. We also study briefly the relationship of our example with the Denjoy-Young-Saks theorem. (C ) 1998 Academic Press.