Functions of two variables with large tangent plane sets
Main result: There exist a \(C^1\) function \(f: \mathbb{R}^2\to \mathbb{R}\), a nowhere dense closed set \(E\subseteq [0,1]\times [0,1]\) of zero two-dimensional measure, and a nonempty open set \(H\subseteq \mathbb{R}^3\) such that for any \((a,b,c)\in H\) we can find an \((x_0,y_0)\in E\) for which the equation of the tangent plane to the surface \(z= f(x,y)\) at \((x_0, y_0)\) is \(z= ax+ by+ c\). As a final remark, the author observes that the Hausdorff dimension of the set \(E\) is less than 2, but not less than 1. Open problem: Can the Hausdorff dimension of \(E\) be equal to 1? It is also proved that the theorem above (whose proof is very laborious) is no longer valid when \(C^1\) is replaced by \(C^2\). The relationship with the Denjoy-Young-Saks theorem (concerning the behavior almost everywhere of the derived numbers of an arbitrary real function of a real variable) is also investigated.
scientific article; zbMATH DE number 1218580
scientific article