{
  "id": "0709.4492",
  "version": "v5",
  "published": "2007-09-27T20:10:46.000Z",
  "updated": "2016-06-30T03:50:15.000Z",
  "title": "Fun with Calculus 1: a dynamical re-analysis of fundamental theorems in calculus",
  "authors": [
    "Daniel Reem"
  ],
  "comment": "20 pages, 3 figures. This version is a major revision which includes, among other things, significant improvements in the presentation (more user friendly), a new title, a new abstract, extensions of various assertions, replacement of the \"envelope proof\" of the extreme value theorem by the \"level-set proof\", and addition of figures, references and acknowledgements",
  "categories": [
    "math.CA",
    "math.HO"
  ],
  "abstract": "Although a first course in calculus is not always a pleasant experience for fresh students, a re-examination of its material using the more sophisticated tools and ways of thinking obtained in later stages can be a real fun for experts, advanced students and many others. Here we try to achieve something in this direction by playing with, and looking for new horizons in, three fundamental theorems in calculus and related issues. We start the trek by going to the extreme, namely discussing the extreme value theorem regarding the extreme (optimal) values of a continuous function defined on a compact space. Two short proofs of this theorem are presented: \"the programmer proof\" that suggests a method (which is practical in down-to-earth settings) to approximate, to any required precision, the extreme values in a metric space setting, and an abstract space proof (\"the level-set proof\") for semicontinuous functions defined on compact topological spaces. In the intermediate part we discuss the intermediate value theorem which is generalized to a class of discontinuous functions, the meaning of the intermediate value property is re-examined, and a fixed point theorem for (very) discontinuous functions is established. The trilogy reached the final frontier in a discussion on uniform continuity in which the optimal delta of the given epsilon is obtained explicitly, several properties of it are derived, and necessary or sufficient conditions for the uniform continuity of a function are established, among them the classical one in which the space is compact and the function is continuous. Have fun!",
  "revisions": [
    {
      "version": "v4",
      "updated": "2010-10-04T15:47:38.000Z",
      "title": "New proofs and improvements of classical theorems in analysis",
      "abstract": "New proofs and improvements of three classical theorems in analysis are presented. Although these theorems are well-known, and have been extensively investigated over the years, it seems that new light can be shed on them. We first present a quantitative necessarily and sufficient condition for a function to be uniformly continuous, and as a by-product we obtain explicitly the optimal delta for the given epsilon. The uniform continuity of a continuous function defined on a compact metric space follows as a simple consequence. We proceed with the extreme value theorem and present a ``programmer's proof'', a proof which does not use the costume argument of proving boundedness first. We finish with the intermediate value theorem, which is generalized to a class of discontinuous functions, and, in addition, the meaning of the intermediate value property is re-examined and a fixed point theorem for (very) discontinuous functions is established. At the end we discuss briefly in which sense the proofs are constructive.",
      "comment": "8 pages; Several changes in the title, abstract, and the introduction section",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2016-06-30T03:50:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03F99",
      "26A15",
      "26A03",
      "47H10",
      "54D05",
      "90C26"
    ],
    "keywords": [
      "classical theorems",
      "improvements",
      "intermediate value theorem",
      "compact metric space",
      "discontinuous functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.4492R"
    }
  }
}