复导数是连续的-三个自包含的证明。第1部分
Complex derivatives are continuous - three self-contained proofs. Part 1
摘要：
我们用三种方法证明了分析的基本事实：复导数是连续的。第一，经典，证明柯西和Goursat使用积分。Whyburn和Connell的第二个证明是拓扑的，并且是基于卷绕数的。第三个证明Adel'son-Vel'skii和Kronrod采用图形的真正二元函数。我们给出了第一个证明的简短而简洁的介绍（在教科书中，它通常超过几十页）。在第二和第三个证明中，我们相反地填充和扩展省略的步骤和辅助结果。第一部分给出了前两个证明。第三个证明，治疗在未来的第2部分，需要一个巡回赛通过定理和结果由于约旦，布劳威尔，Urysohn，Tietze，维塔利，萨尔德，富比尼，Komarevsky，年轻，柯西，黎曼和其他人。
英文摘要：
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.
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