{
  "id": "2107.00619",
  "version": "v1",
  "published": "2021-07-01T17:15:11.000Z",
  "updated": "2021-07-01T17:15:11.000Z",
  "title": "Cutting sets of continuous functions on the unit interval",
  "authors": [
    "Marek Balcerzak",
    "Piotr Nowakowski",
    "Michał Popławski"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "For a function $f\\colon [0,1]\\to\\mathbb R$, we consider the set $E(f)$ of points at which $f$ cuts the real axis. Given $f\\colon [0,1]\\to\\mathbb R$ and a Cantor set $D\\subset [0,1]$ with $\\{0,1\\}\\subset D$, we obtain conditions equivalent to the conjunction $f\\in C[0,1]$ (or $f\\in C^\\infty [0,1]$) and $D\\subset E(f)$. This generalizes some ideas of Zabeti. We observe that, if $f$ is continuous, then $E(f)$ is a closed nowhere dense subset of $f^{-1}[\\{ 0\\}]$ where each $x\\in \\{0,1\\}\\cap E(f)$ is an accumulation point of $E(f)$. Our main result states that, for a closed nowhere dense set $F\\subset [0,1]$ with each $x\\in \\{0,1\\}\\cap E(f)$ being an accumulation point of $F$, there exists $f\\in C^\\infty [0,1]$ such that $F=E(f)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-01T17:15:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C30",
      "54E52",
      "26A30",
      "26A24"
    ],
    "keywords": [
      "unit interval",
      "cutting sets",
      "continuous functions",
      "accumulation point",
      "main result states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}