Ordinary and strong density continuity of complex analytic functions

Abstract

In the paper we prove that the complex analytic functions are (ordinarily) density continuous. This stays in contrast with the fact that even such a simple function as G: R 2 --~ R 2, G(x, y) = (x, y3), is not density continuous [1]. We will also characterize those analytic functions which are strongly density continuous at the given point a E C. From this we conclude that a complex analytic function f is strongly density continuous if and only if f(z) = a + bz, where a, b E C and b is either real or imaginary. 1. Preliminaries The notation used throughout this paper is standard. In particular, the complex plane C will be identified with R 2. All sets considered in the paper will be Lebesgue measurable. The two-dimensional Lebesgue measure of a set A C C will be denoted by ~(A). Recall that 0 is a strong dispersion point of A C C if