{
  "id": "1805.05819",
  "version": "v1",
  "published": "2018-05-15T14:41:18.000Z",
  "updated": "2018-05-15T14:41:18.000Z",
  "title": "Two notes on generalized Darboux properties and related features of additive functions",
  "authors": [
    "Gabriel Istrate"
  ],
  "comment": "This is a paper from a special issue dedicated to the 90th birthday of Professor Solomon Marcus. Since the journal is not available online/indexed as of 2018, I am placing a copy of the paper here",
  "journal": "Annals of the University of Bucharest Informatics Series, Anul LXII, no. 2 (2015), 61-77",
  "categories": [
    "math.CA"
  ],
  "abstract": "We present two results on generalized Darboux properties of additive real functions. The first results deals with a weak continuity property, called ${\\bf Q}$-continuity, shared by all additive functions. We show that every ${\\bf Q}$-continuous function is the uniform limit of a sequence of Darboux functions. The class of ${\\bf Q}$-continuous functions includes the class of Jensen convex functions. We discuss further connections with related concepts, such as ${\\bf Q}$-differentiability. Next, given a ${\\bf Q}$-vector space $A\\subseteq {\\bf R}$ of cardinality ${\\bf c}$ we consider the class ${\\cal DH}^{*}(A)$ of additive functions such that for every interval $I\\subseteq {\\bf R}$, $f(I)=A$. We show that every function in class ${\\cal DH}^{*}(A)$ can be written as the sum of a linear (additive continuous) function and an additive function with the Darboux property if and only if $A={\\bf R}$. We apply this result to obtain a relativization of a certain hierarchy of real functions to the class of additive functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-15T14:41:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A15"
    ],
    "keywords": [
      "additive function",
      "generalized darboux properties",
      "darboux property",
      "related features",
      "continuous function"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}