Discussion Overview
The discussion revolves around the concept of differentiability of functions in closed and open intervals, specifically exploring whether a function can be differentiable a certain number of times in a closed interval and more times in an open interval. The conversation references Taylor's theorem and examines specific examples to illustrate the points raised.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant questions the possibility of a function being differentiable n times in a closed interval and (n+1) times in an open interval, suggesting it relates to differentiability at endpoints.
- Another participant provides the example of the function ##f(x) = x^{4/3}##, noting it is twice differentiable on the open interval ##(0,1)## but only once at the endpoint ##x=0##.
- Some participants express confusion regarding the differentiability of the function at the endpoints, specifically questioning whether it is twice differentiable in the closed interval ##[0,1]##.
- There is a discussion about the second derivative of the function, with one participant stating it as ##f(x)=4/9x^{-2/3}## and recognizing that it cannot be defined at ##x=0##, which contributes to the function's differentiability status.
Areas of Agreement / Disagreement
Participants exhibit uncertainty regarding the differentiability of the function at the endpoints, with some agreeing on the limitations of differentiability at those points while others seek clarification on the definitions and implications.
Contextual Notes
There are unresolved questions about the definitions of differentiability at endpoints and the implications of the second derivative's behavior at those points. The discussion also highlights the importance of limits from both sides in determining differentiability.