Holomorphy of an arbitrary approximately holomorphic mapping from a plane domain into the plane (Q1334443)

The author proves a sufficient condition on a (generally discontinuous) mapping \(f: D\mapsto \mathbb{C}\), where \(D\) is a domain in the complex plane \(\mathbb{C}\), to be holomorphic in \(D\) based on the notion of an approximate derivative. Definition: The approximate derivative \(f_{ap}'(z_ 0)\) of \(f\) at a point \(z_ 0\in D\) is the limit \[ \lim_{{\begin{smallmatrix} z\to z_ 0\\ z\in E(z_ 0)\end{smallmatrix}}} {f(z)- f(z_ 0)\over z- z_ 0}, \] if the limit exists, where \(E(z_ 0)\) is a set in \(D\) such that \[ \lim_{r\to 0} {m^*(CE(z_ 0)\cap C(z_ 0, r))\over \pi r^ 2} = 0; \] here \(CE(z_ 0)\) denotes the complement of \(E(z_ 0)\) to the domain \(D\), and \(C(z_ 0, r)\) denotes the open disk about \(z_ 0\) of radius \(r\). The author proves Theorem 2: Let \(f: D\mapsto \mathbb{C}\) be a one-to-one mapping with finite approximate derivative at each point of \(D\). Then the mapping is necessarily holomorphic in \(D\). This theorem is an analogue of a similar result in which continuity is assumed that bijectivity is not. The proof is rather lengthy. It relies on the following five results. (a) If \(f\) has a finite approximate derivative on a perfect subset of \(D\) then \(f\) satisfies the Lipschitz condition on a portion of that set. (b) There exists an everywhere dense open subset of \(D\) in which \(f\) is holomorphic. (c) \(f\) is absolutely continuous on almost all sections by parallel lines to the coordinate axes of a rectangle whose diameter is sufficiently small and whose sides are parallel to the coordinate axes. (d) \(f(z)\) satisfies the Cauchy-Riemann equations almost everywhere in a certain small rectangle in \(D\). (e) \(\oint f(z) dz= 0\) in rectangles as in (d) and hence \(f\) is holomorphic.