{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0PAQM3H57X"
    },
    {
      "field": "uses",
      "target": "0T4EJ8ZJM9"
    }
  ],
  "body": "\n## Definition\n\nLet $D \\subseteq \\mathbb{R}$ and let $f \\colon D \\to \\mathbb{R}$ be a function.\nThe function $f$ is sequentially continuous if the following holds.\nFor each sequence $(x_n)$ in $D$ that [[converges|0XKWSSY8P2]] to a point $a \\in D$, the sequence $(f(x_n))$ converges to $f(a)$.\n\n## Remarks\n\nFor real functions, [[sequential continuity is the same as continuity|0PAQM3H57X]].\nThe definition is useful to prove that a function is not continuous. It is enough to find one sequence $x_n \\to a$ with $f(x_n) \\not\\to f(a)$.\nMathlib defines it for topological spaces, as `SeqContinuous`.\n\n## Explanations\n\n### L1 human\n\nA sequentially continuous function preserves limits of sequences.\nThe function of the limit is the limit of the function values.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "055Z15H69G",
    "kind": "definition",
    "msc": [
      "26A15"
    ],
    "name": "Sequential continuity",
    "sources": [
      {
        "locator": "Definition in terms of limits of sequences",
        "url": "https://en.wikipedia.org/wiki/Continuous_function"
      }
    ],
    "tag": "0MVDQEAB60",
    "uses": [
      "0XKWSSY8P2"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0MVDQEAB60/",
  "links": [
    {
      "field": "defines",
      "target": "055Z15H69G"
    },
    {
      "field": "uses",
      "target": "0XKWSSY8P2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0MVDQEAB60.md",
  "msc": [
    "26A15"
  ],
  "name": "Sequential continuity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0MV/0MVDQEAB60.md",
  "status": null,
  "tag": "0MVDQEAB60",
  "uri": "m2://0MVDQEAB60",
  "url": "https://math2.tamnd.com/wiki/sequential-continuity-definition/",
  "writing": {
    "tag": "0MVDQEAB60",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}