Functions of Two Variables with Large Tangent Plane Sets

We show that there exist a C1 function, f; of two variables and a set E R2 of zero Lebesgue measure such that using the natural three-dimensional parametrization of planes z D ax C by C c tangent to the surface z D f x; y‘, the (threedimensional) 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 C2 functions. We also study briefly the relationship of our example with the Denjoy–Young–Saks theorem.