Discussion Overview
The discussion revolves around the equicontinuity of the family of functions defined by F_n(x) = sin(nx) and F_n(x) = cos(nx). Participants explore the definition of equicontinuity and attempt to prove or disprove whether these families of functions are equicontinuous at specific points, particularly at x_0 = 0.
Discussion Character
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant attempts to prove equicontinuity of F_n(x) = sin(nx) at x_0 = 0 using the definition of equicontinuity, noting that delta depends on epsilon and x.
- Another participant corrects the definition of equicontinuity provided by the first participant, stating that it should indicate that for all ε > 0, there exists δ > 0 such that if d(x_0, x) < δ, then d(f(x_0), f(x)) < ε.
- This second participant argues that if the family {sin(nx)} is equicontinuous, then for ε = 1/2, there should exist δ such that if |x| < δ, then |sin(nx)| < ε for all n. They provide a counterexample to show that this leads to a contradiction, suggesting that the family cannot be equicontinuous.
- A third participant reiterates the correction regarding the definition of equicontinuity, emphasizing the need for the condition to hold for all n.
- Another participant proposes a similar argument for the family F_n(x) = cos(nx), suggesting that it is also not equicontinuous by following a similar reasoning process and providing a specific counterexample.
Areas of Agreement / Disagreement
Participants generally disagree on the equicontinuity of the families of functions. While some argue that neither family is equicontinuous, the discussion remains unresolved regarding the implications of the definitions and the validity of the proofs presented.
Contextual Notes
Participants express uncertainty regarding the application of the definition of equicontinuity and the implications of their counterexamples. The discussion highlights the need for careful consideration of the conditions under which equicontinuity is evaluated.