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166: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/06/12(木) 22:34:28.84 ID:EWvjXceg

つづき

Proof Outline:
1. Definition of Extension:
The extension F is defined on X by considering a sequence {x_n} in S that converges to x in X. Then F(x) is defined as the limit of f(x_n) as n approaches infinity.
2. Well-Definedness:
The definition of F is shown to be independent of the chosen sequence {x_n} converging to x.
3. Continuity of Extension:
The extension F is shown to be continuous on the closure of S (i.e., ¯S).
4. Uniform Continuity of Extension:
The uniform continuity of F is established using the uniform continuity of f and the completeness of Y.

Significance:
This theorem is fundamental in analysis because it allows us to extend properties of functions defined on dense subsets to the entire space. This is particularly useful when we want to analyze the behavior of functions on a larger space using information available on a smaller, dense subset

(参考リンク：URL略す)
Extending a function by continuity from a dense subset of a space
2011/10/29 — Now, the main theorem. Theorem. Let X and Y be metric spaces, S a subset of X, and f:S→Y. If f is uniformly continuous a...
Mathematics Stack Exchange

Continuous extensions of continuous functions on dense subspaces
2012/07/12 — 1 Answer. ... Uniform continuity ensures that the Cauchy sequence (qn) in Q is mapped to a Cauchy (and hence convergent)
Mathematics Stack Exchange

Uniform continuity - Wikipedia
Continuity of a function for metric spaces and at every point of an interval (i.e., continuity of on the interval ) is expressed b...
Wikipedia, the free encyclopedia
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