Directional Differentiability in the Euclidean Plane

Abstract

Smoothness conditions on a function f : R 2 → R that are weaker than being differentiable or Lipschitz at a point are defined and studied.

A note on directional Lipschitz continuity in the Euclidean plane

“…In order to put this statement in context, consider the following heuristic principle (cf. also the discussion concluding the first section of [1]):…”

“…If f : R d → R is any function and E denotes the set of points where f is smooth in many directions, then f is Lipschitz relative to E on a large subset of E (or R d ). (1) The main result of [1], Theorem 4, disproves a variant of this principle with d = 2 and the emphasized parts replaced by differentiable, almost every direction and a subset of E of full measure, respectively. Conjecture 1 then concerns with another precise version of (1).…”

“…(1) The main result of [1], Theorem 4, disproves a variant of this principle with d = 2 and the emphasized parts replaced by differentiable, almost every direction and a subset of E of full measure, respectively. Conjecture 1 then concerns with another precise version of (1). This paper contributes to investigation of the principle (1) in two ways.…”

“…Our construction further exploits the ideas introduced in the proof of Theorem 4 from [1] and, in addition, employs an iterative argument. We start by stating two useful facts proven in [1]. Proposition 2.…”

“…We proceed in three steps. The first step generalizes the construction from Paragraph 3.2 of [1] to arbitrary open subset of R 2 and defines the set Z by iterating this construction. In the second step we define the "good" set E and prove that it has full measure in R 2 .…”