Cauchy-Riemann Equations and $\mathbb{C}$-differentiable Functions

Abstract

The definitions of limit and continuity depend really on the metric space structure of C. All the usual results on continuity of sums, products, quotient and composition still hold here, and we will not recall them. These are local properties. The specific structure of C, or of its subsets, will come into play when one studies the existence of a continuous function in a given set.