  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.
