{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe prove each direction separately. Both metric spaces carry their metric topologies.\n\n1. Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]], let $f \\colon X \\to Y$ and let $a \\in X$.\n2. Write $B_X(a, \\delta)$ and $B_Y(f(a), \\varepsilon)$ for the [[open balls|0DC3MPR7C9]] of $X$ and $Y$. By the symmetry of a metric, $x \\in B_X(a, \\delta)$ if and only if $d_X(x, a) \u003c \\delta$.\n3. Recall that a set $N$ is a [[neighbourhood|09CJ00H3NW]] of a point $p$ when some open set $U$ satisfies $p \\in U \\subseteq N$.\n4. First suppose that $f$ is [[continuous at|07N7W5C29X]] $a$. Let $\\varepsilon \u003e 0$ and put $V = B_Y(f(a), \\varepsilon)$.\n5. By [[open balls are open|0B2305GNAZ]], $V$ is open. Also $f(a) \\in V$, as $d_Y(f(a), f(a)) = 0 \u003c \\varepsilon$. So $V$ is a neighbourhood of $f(a)$.\n6. By continuity at $a$, the [[preimage|0FGCFY1AZX]] $f^{-1}(V)$ is a neighbourhood of $a$. So there is an open set $U$ with $a \\in U \\subseteq f^{-1}(V)$.\n7. By the [[description of the open sets of a metric space|0A26X79RTB]], there is $\\delta \u003e 0$ with $B_X(a, \\delta) \\subseteq U$.\n8. Let $x \\in X$ with $d_X(x, a) \u003c \\delta$. By step 2, $x \\in B_X(a, \\delta) \\subseteq f^{-1}(V)$, so $f(x) \\in V$.\n9. Thus $d_Y(f(a), f(x)) \u003c \\varepsilon$, and by symmetry $d_Y(f(x), f(a)) \u003c \\varepsilon$. This is the epsilon-delta condition.\n10. Conversely, suppose the epsilon-delta condition holds at $a$. Let $V$ be a neighbourhood of $f(a)$. Choose an open set $W$ with $f(a) \\in W \\subseteq V$.\n11. By the description of open sets, there is $\\varepsilon \u003e 0$ with $B_Y(f(a), \\varepsilon) \\subseteq W$. Take $\\delta \u003e 0$ as given by the condition for this $\\varepsilon$.\n12. Let $x \\in B_X(a, \\delta)$. By step 2, $d_X(x, a) \u003c \\delta$, so $d_Y(f(x), f(a)) \u003c \\varepsilon$.\n13. By symmetry, $f(x) \\in B_Y(f(a), \\varepsilon) \\subseteq V$. Hence $B_X(a, \\delta) \\subseteq f^{-1}(V)$.\n14. The ball $B_X(a, \\delta)$ is open, because open balls are open, and it contains $a$. So $f^{-1}(V)$ is a neighbourhood of $a$.\n15. As $V$ was arbitrary, $f$ is continuous at $a$.\n\n## Idea\n\nIn a metric space, the open balls about a point are small enough to stand in for all its neighbourhoods. Every neighbourhood contains a ball, and every ball is a neighbourhood. So the condition on neighbourhoods becomes a condition on balls, which is the epsilon-delta condition.\n",
  "commit": "68a8c3dc73fd55004b2796431c607c30960d1a85",
  "front": {
    "kind": "proof",
    "msc": [
      "54E35"
    ],
    "name": "Proof of epsilon-delta continuity at a point in metric spaces",
    "proves": "0EKGKY5TGH",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Continuous_function"
      }
    ],
    "tag": "G7B6S4CKNF",
    "uses": [
      "063W5JMY85",
      "07N7W5C29X",
      "09CJ00H3NW",
      "0FGCFY1AZX",
      "0DC3MPR7C9",
      "0B2305GNAZ",
      "0A26X79RTB"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/G7B6S4CKNF/",
  "links": [
    {
      "field": "proves",
      "target": "0EKGKY5TGH"
    },
    {
      "field": "uses",
      "target": "063W5JMY85"
    },
    {
      "field": "uses",
      "target": "07N7W5C29X"
    },
    {
      "field": "uses",
      "target": "09CJ00H3NW"
    },
    {
      "field": "uses",
      "target": "0FGCFY1AZX"
    },
    {
      "field": "uses",
      "target": "0DC3MPR7C9"
    },
    {
      "field": "uses",
      "target": "0B2305GNAZ"
    },
    {
      "field": "uses",
      "target": "0A26X79RTB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/G7B6S4CKNF.md",
  "msc": [
    "54E35"
  ],
  "name": "Proof of epsilon-delta continuity at a point in metric spaces",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/G7B/G7B6S4CKNF.md",
  "status": null,
  "tag": "G7B6S4CKNF",
  "uri": "m2://G7B6S4CKNF",
  "url": "https://math2.tamnd.com/wiki/proof-of-epsilon-delta-continuity-at-a-point-in-metric-spaces/",
  "writing": {
    "tag": "G7B6S4CKNF",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}