{
  "id": "2007.12752",
  "version": "v1",
  "published": "2020-07-24T19:44:44.000Z",
  "updated": "2020-07-24T19:44:44.000Z",
  "title": "On the $L^1$ and pointwise divergence of continuous functions",
  "authors": [
    "Karol Gryszka",
    "Paweł Pasteczka"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "For a family of continuous functions $f_1,f_2,\\dots \\colon I \\to \\mathbb{R}$ ($I$ is a fixed interval) with $f_1\\le f_2\\le \\dots$ define a set $$ I_f:=\\big\\{x \\in I \\colon \\lim_{n \\to \\infty} f_n(x)=+\\infty\\big\\}.$$ We study the properties of the family of all admissible $I_f$-s and the family of all admissible $I_f$-s under the additional assumption $$ \\lim_{n \\to \\infty} \\int_x^y f_n(t)\\:dt=+\\infty \\quad \\text{ for all }x,y \\in I\\text{ with }x<y.$$ The origin of this problem is the limit behaviour of quasiarithmetic means.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-24T19:44:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26E60",
      "54C30",
      "26A15",
      "26A30",
      "28A78"
    ],
    "keywords": [
      "continuous functions",
      "pointwise divergence",
      "additional assumption",
      "limit behaviour",
      "quasiarithmetic means"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}