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    "title": "Tietze extends bounded continuous functions from closed subsets of metric spaces",
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        "title": "Über Funktionen, die auf einer abgeschlossenen Menge stetig sind (JReineAngewMath145,9–14; GDZscans)",
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        "description": "The original displayed extension theorem assumes a bounded real-valued continuous function and a closed subset of a metric space. It is not presented here as the later general equivalence for every normal topological space, an unbounded/vector-valued extension theorem, a unique extension,or automatic preservation of differentiability or Lipschitz constants. No completeness or separability assumption is silently added. Modern textbook generalizations and different explicit extension formulas are not backdated to this source.",
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          "citation": "Über Funktionen, die auf einer abgeschlossenen Menge stetig sind (JReineAngewMath145,9–14; GDZscans)",
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      "statement": "# Tietze extends bounded continuous functions from closed subsets of metric spaces\n\nTietze proves that a bounded real-valued continuous function on a closed subset of a Frechet classE—a metric space in the paper's explicit distance axioms—extends continuously to the whole space. The construction uses distance to the closed set and a distance-weighted upper envelope. Related majorization and decreasing continuous approximation results supply the same proof framework.\n\n## Scientific scope\n\nThe original displayed extension theorem assumes a bounded real-valued continuous function and a closed subset of a metric space. It is not presented here as the later general equivalence for every normal topological space, an unbounded/vector-valued extension theorem, a unique extension,or automatic preservation of differentiability or Lipschitz constants. No completeness or separability assumption is silently added. Modern textbook generalizations and different explicit extension formulas are not backdated to this source.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nHeinrich Tietze\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [Über Funktionen, die auf einer abgeschlossenen Menge stetig sind (JReineAngewMath145,9–14; GDZscans)](https://gdz.sub.uni-goettingen.de/download/pdf/PPN243919689_0145/LOG_0006.pdf)\n",
      "summary": "Tietze proves that a bounded real-valued continuous function on a closed subset of a Frechet classE—a metric space in the paper's explicit distance axioms—extends continuously to the whole space. The construction uses distance to the closed set and a distance-weighted upper envelope. Related majorization and decreasing continuous approximation results supply the same proof framework.",
      "title": "Tietze extends bounded continuous functions from closed subsets of metric spaces"
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    "id": "tietze-bounded-continuous-closed-metric-extension-1915",
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    "title": "Tietze extends bounded continuous functions from closed subsets of metric spaces",
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    "what_changed": "Tietze proves that a bounded real-valued continuous function on a closed subset of a Frechet classE—a metric space in the paper's explicit distance axioms—extends continuously to the whole space. The construction uses distance to the closed set and a distance-weighted upper envelope. Related majorization and decreasing continuous approximation results supply the same proof framework.",
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