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A CONTINUOUS TALE ON CONTINUOUS AND SEPARATELY CONTINUOUS FUNCTIONS

机译：连续函数和连续函数的连续故事

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摘要

Assume that you have developed a good set of tools allowing you to decide which real functions of one real variable, f: R → R, are continuous. (Q): How can such a toolbox be utilized to decide on the continuity of the functions g: R~n→ R of n real variables ? This is one of the questions which must be faced by any student taking multivariable calculus. Of course, such a student is following the footsteps of many generations of mathematicians, which were, and still are, struggling with the same general question. The aim of this article is to present the history and the current research related to this subject in a real analysis perspective, rather than in a more general, topological perspective. In addition to surveying the results published so far, this exposition includes several original results (Theorems 1, 12 and 13), as well as some new simplified versions of the (sketches of the) proofs of older results. We also recall several intriguing open problems.
机译：假设您已经开发了一套好的工具，可以确定一个实变量f：R→R的哪些实函数是连续的。 （Q）：如何利用这样的工具箱来确定函数g的连续性：n个实变量的R〜n→R？这是参加多变量演算的任何学生都必须面对的问题之一。当然，这样的学生遵循着几代数学家的脚步，他们曾经并且仍然在为同样的一般问题而苦苦挣扎。本文的目的是从真实的分析角度而不是从更一般的拓扑角度介绍与该主题相关的历史和当前研究。除了调查到目前为止已发布的结果外，该博览会还包括一些原始结果（定理1、12和13），以及一些较旧的结果证明的新简化版本。我们还记得几个有趣的开放问题。

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  • 来源
    《Real analysis exchange》 |2016年第1期|19-54|共36页
  • 作者

    Krzysztof Chris Ciesielski; David Miller;

  • 作者单位

    Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310 Department of Radiology, MIPG, University of Pennsylvania, Blockley Hall-4th Floor, 423 Guardian Drive, Philadelphia, PA 19104-6021;

    Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310;

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  • 原文格式 PDF
  • 正文语种 eng
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  • 关键词

    separate continuity; linear continuity;

    机译：单独的连续性;线性连续性;

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