{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-continuity",
  "semantic_family_id": "sf-continuity",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實函數的點值與連續"
  },
  "summary": {
    "zh-Hant": "在全實數論域連續需要點值存在且與兩側極限一致；補點與原函數連續不可混用。"
  },
  "statement": {
    "zh-Hant": "對f:R→R，f在a連續指每個ε>0都有δ>0，使 |x−a|<δ時 |f(x)−f(a)|<ε。等價地去心兩側極限等於已定義點值f(a)。本輪兩側仿射、中心明示值V的模型，在a連續當且僅當左極限=右極限=V。"
  },
  "formula": {
    "tex": "C(f,a)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        },
        {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "bc6fdb902ae398420a330eb03f94ec7a9c69e6328864e025d0f97abf032ed9bb"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "全實數論域及中心點值已明示，不把未定義中心直接稱作此論域上的不連續點。"
    },
    {
      "id": "a2",
      "zh-Hant": "仿射點值題精確算ma+b；分支題分別算left_m*a+left_b及right_m*a+right_b，中心值V獨立給定。"
    },
    {
      "id": "a3",
      "zh-Hant": "補定可去商的中心值以形成全R連續延拓時，唯一所需值為2a；補點前原商仍只在R∖{a}有定義。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "continuity_at_point",
      "role_zh": "全實數函數在一點連續的述詞",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "real_total_function",
      "role_zh": "指定全實數總函數或補點後目標",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "continuity_point",
      "role_zh": "明示實數中心",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-function-limit",
      "title_zh": "去心實數論域的函數極限",
      "reason_zh": "先核對兩側極限。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "點值是總函數宣告的一部分。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "左右極限相等仍不足以連續，還須等於中心值；中心值只等於其中一側但另一側不同也不連續。",
      "補點題問的是如何定義新的全實數延拓，不把原本去心商已經當成有中心值的總函數。可去不等於不需補定。",
      "點缺陷對 (左極限−V,右極限−V) 精確顯示失配；二者都零才符合本模型的中心連續。共同平移全部輸出不產生新缺陷計算，改旁側斜率卻保持兩側極限也不充量。"
    ],
    "common_misconception_zh": "左右極限相等就忽略點值，或未補定便聲稱原商在中心連續。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "明示分支與補點條件可重算；一般連續性理論與形式 Evidence 尚未建立。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "在固定版綁定 ContinuousAt、full real domain與補點延拓，不混用去心極限及未定義點。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-continuity.md 整理；2026-10-07 脈衝 R03（使用者手動接續 2026-10-08 工作序列；實際日期仍為 2026-10-07），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-07"
  }
}