{
  "id": "math/9811011",
  "version": "v1",
  "published": "1998-11-03T17:18:16.000Z",
  "updated": "1998-11-03T17:18:16.000Z",
  "title": "Two examples concerning almost continuous functions",
  "authors": [
    "Krzysztof Ciesielski",
    "Andrzej Roslanowski"
  ],
  "comment": "22 pages",
  "journal": "Topology Appl. 103 (2000) 187-202",
  "categories": [
    "math.CA",
    "math.LO"
  ],
  "abstract": "We construct, under the assumption that union of less than continuum many meager subsets of R is meager in R, an additive connectivity function f:R-->R with Cantor intermediate value property which is not almost continuous. This gives a partial answer to a question of D. Banaszewski. We also show that every extendable function g:R-->R with a dense graph satisfies the following stronger version of the SCIVP property: for every a<b and every perfect set K between g(a) and g(b) there is a perfect subset C of (a,b) such that g[C] subset K and g|C is continuous strictly increasing. This property is used to construct a ZFC example of an additive almost continuous function f:R-->R which has the strong Cantor intermediate value property but is not extendable. This answers a question of H. Rosen. This also generalizes Rosen's result that a similar (but not additive) function exists under the assumption of the continuum hypothesis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1998-11-03T17:18:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A15",
      "26A30",
      "03E50"
    ],
    "keywords": [
      "continuous functions",
      "examples concerning",
      "strong cantor intermediate value property",
      "generalizes rosens result",
      "dense graph satisfies"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1998math.....11011C"
    }
  }
}