Complex derivatives are continuous - three self-contained proofs. Part 1

TL;DR

This paper presents three distinct proofs demonstrating that complex derivatives are continuous, including classical, topological, and graph-based approaches, with detailed explanations and auxiliary results.

Contribution

It provides three self-contained proofs of the continuity of complex derivatives, expanding on classical and modern methods with detailed explanations.

Findings

Classical proof using integration confirms continuity.

Topological proof based on winding number establishes continuity.

Graph-based proof employs real bivariate function graphs.

Abstract

We prove in three ways the basic fact of analysis that complex derivatives are continuous. The first, classical, proof of Cauchy and Goursat uses integration. The second proof of Whyburn and Connell is topological and is based on the winding number. The third proof of Adel'son-Vel'skii and Kronrod employs graphs of real bivariate functions. We give short and concise presentation of the first proof (in textbooks it often spreads over dozens of pages). In the second and third proof we on the contrary fill in and expand omitted steps and auxiliary results. This part 1 presents the first two proofs. The third proof, treated in the future part 2, takes one on a tour through theorems and results due to Jordan, Brouwer, Urysohn, Tietze, Vitali, Sard, Fubini, Komarevsky, Young, Cauchy, Riemann and others.