{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-uniform-limit-continuity-theorem",
  "semantic_family_id": "sf-uniform-limit-continuity-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "一致極限保持連續"
  },
  "summary": {
    "zh-Hant": "同一domain上的連續函數列若一致收斂至f，則極限函數f連續。"
  },
  "statement": {
    "zh-Hant": "在共同domain上，每個f_n連續且sup_x |f_n(x)-f(x)|→0。題族用[0,1]上的affine perturbation、exact sup error與epsilon-N證書重算，並以epsilon/3 transfer把一個固定f_N的continuity傳給f。pointwise-only的x^n、uniform收斂到discontinuous limit但各項不連續、以及有限sample closeness都明示為失敗邊界。"
  },
  "formula": {
    "tex": "U(F,f,e)=b",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>U</mi><mo>(</mo><mi>F</mi><mo>,</mo><mi>f</mi><mo>,</mo><mi>e</mi><mo>)</mo></mrow><mo>=</mo><mi>b</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "U",
          "glyph": "U"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "F",
            "glyph": "F"
          },
          {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          },
          {
            "type": "symbol",
            "name": "e",
            "glyph": "e"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "b",
        "glyph": "b"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ac1cc0d74035bc38d13ea8ec605f607f75edf212db48eab5a4073abed1910d75"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "全部函數共享明示domain與codomain。"
    },
    {
      "id": "a2",
      "zh-Hant": "uniform convergence的N只依epsilon，不依x；每個approximant的continuity另行成立。"
    }
  ],
  "symbols": [
    {
      "token": "U",
      "role": "uniform_limit_operator",
      "role_zh": "一致極限定理算子",
      "scope": "representative_formula"
    },
    {
      "token": "F",
      "role": "continuous_function_sequence",
      "role_zh": "連續函數列",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "uniform_limit_function",
      "role_zh": "一致極限函數",
      "scope": "representative_formula"
    },
    {
      "token": "e",
      "role": "positive_tolerance",
      "role_zh": "正容許誤差",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "continuity_conclusion",
      "role_zh": "連續性結論",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-uniform-convergence-of-functions",
      "title_zh": "函數列一致收斂",
      "reason_zh": "核心收斂假設。"
    },
    {
      "id": "mko-continuity",
      "title_zh": "連續性",
      "reason_zh": "逐項與極限結論。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "共同domain/codomain。"
    },
    {
      "id": "mko-supremum-and-infimum",
      "title_zh": "上確界與下確界",
      "reason_zh": "整個domain上的sup error。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "sup error與epsilon-N量詞先於結論。",
      "epsilon/3 transfer固定一個continuous approximant，不要求共同delta。",
      "pointwise convergence與finite samples皆不足。"
    ],
    "common_misconception_zh": "把逐點收斂或有限網格接近當成一致收斂，或漏掉每個f_n連續。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "affine exact families只核量詞與bound；一般拓撲／度量版本仍需證明。",
    "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": "加入uniform convergence下積分交換與導數交換的額外條件。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-uniform-limit-continuity-theorem.md 整理；2026-10-15 脈衝 R03（使用者手動接續 2026-11-17 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}