Functions of two 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 corresponding to points (x, y) in E is nonempty. From the Morse-Sard theorem it 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.