Discussion Overview
The discussion revolves around the relationship between the differentiability of a function of two variables and the equality of mixed partial derivatives, specifically whether double differentiability implies the equality of the mixed partial derivatives fxy and fyx.
Discussion Character
- Technical explanation, Conceptual clarification, Debate/contested
Main Points Raised
- Some participants propose that double differentiability of a function implies the equality of mixed partial derivatives fxy and fyx, contingent on the continuity of the second derivatives.
- Others argue that while double differentiability is a strong condition, it is not sufficient alone to establish the equality of mixed partials without the continuity of the second derivatives at the point of interest.
- A later reply suggests that the equality of cross partials is generally true unless the function is pathological, indicating that typical functions will satisfy this condition.
Areas of Agreement / Disagreement
Participants express differing views on the implications of differentiability and continuity regarding the equality of mixed partial derivatives, indicating that the discussion remains unresolved with multiple competing perspectives.
Contextual Notes
Limitations include the need for clarity on the definitions of differentiability and continuity in the context of mixed partial derivatives, as well as the potential for pathological cases that may not conform to general expectations.