Discussion Overview
The discussion revolves around the proposition that every continuous function defined on an unbounded set is uniformly continuous. Participants explore the implications of this claim, referencing examples and counterexamples to clarify their positions.
Discussion Character
- Debate/contested
- Technical explanation
- Conceptual clarification
Main Points Raised
- One participant proposes that if a set \( E \subset \mathbb{R} \) is unbounded, then every continuous function \( f \) defined on \( E \) is uniformly continuous.
- Another participant questions this proposition, suggesting that it should be bounded instead of unbounded, and notes that for uniform continuity, \( E \) must also be closed.
- Several participants point out that the example of \( \mathbb{Z} \) is not applicable because \( \mathbb{Z} \) is discrete, and all functions on discrete sets are uniformly continuous.
- There is a discussion about the definition of unbounded sets, with one participant asserting that if \( E = \mathbb{R} \), then all continuous functions are uniformly continuous.
- Another participant references a theorem from Rudin, indicating that there exist continuous functions on unbounded sets that are not uniformly continuous, emphasizing the importance of boundedness in the theorem's hypotheses.
- A participant provides a specific example of a function defined on the rational numbers that is not uniformly continuous, highlighting that unbounded sets can lead to such functions.
Areas of Agreement / Disagreement
Participants do not reach a consensus; multiple competing views remain regarding the relationship between continuity and uniform continuity on unbounded sets.
Contextual Notes
There are references to specific theorems and examples that illustrate the complexity of the topic, including the distinction between bounded and unbounded sets and the properties of discrete sets.
Who May Find This Useful
This discussion may be of interest to those studying real analysis, particularly in understanding the nuances of continuity and uniform continuity in relation to the properties of sets.