AWhat 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.
Scientific scope
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. 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.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. 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.
Contributors
- The named authors in the research record · SCIENCE: Source-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.
- Über Funktionen, die auf einer abgeschlossenen Menge stetig sind (JReineAngewMath145,9–14; GDZscans)
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