Discussion Overview
The discussion revolves around the definitions and implications of being infinitely differentiable in the context of mathematical functions. Participants explore the nuances between being infinitely many times continuously differentiable and simply infinitely many times differentiable, particularly in relation to the concept of C^∞ functions.
Discussion Character
- Conceptual clarification
- Debate/contested
Main Points Raised
- Some participants suggest that "infinitely many times continuously differentiable" and "infinitely many times differentiable" may not be equivalent, raising questions about the definition of C^∞.
- One participant notes that being infinitely differentiable refers to a property at a specific point, emphasizing that the continuity of the (n-1)th derivative is necessary to obtain the nth derivative at that point.
- Another participant points out that the operation of differentiating a function "infinitely many times" is considered undefined, clarifying that an "infinitely differentiable" function is one for which the Nth derivative exists for any finite integer N > 0.
- A participant states that a differentiable function must be continuous.
Areas of Agreement / Disagreement
Participants express differing views on the equivalence of the terms related to infinite differentiability, indicating that the discussion remains unresolved with multiple competing interpretations.
Contextual Notes
There are limitations in the discussion regarding the definitions and assumptions surrounding differentiability and continuity, particularly in the context of infinite operations and their implications.