Directional differentiability in the Euclidean plane (Q1704531)

In the present work the authors define and study smoothness conditions on a function \(f:\mathbb{R}^2\to \mathbb{R}\) that are weaker than the condition of this function to be differentiable or Lipschitz at a point in the plane. More precisely, the main result is : There is a function \(\chi:\mathbb{R}^2\to \mathbb{R}\) and a set \(E\subset\mathbb{R}^2\) of positive measures such that \(\chi \) is differentiable in almost every direction at each point of \(E\), but \(\chi \) is not Lipschitz a.e. on \(E\).