Not possible to find a function g such that g

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Discussion Overview

The discussion centers around the properties of functions in relation to differentiability and continuous differentiability (C1) at the point (0,0). Participants explore the implications of these definitions and provide examples to illustrate their points.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question the possibility of finding a function g that is continuously differentiable at (0,0) but not differentiable at that point, suggesting that the definitions of C1 and differentiability are inherently contradictory.
  • Another participant presents the function f(x,y) = x|y| as a case study to demonstrate that it is not in C1 at (0,0), implying that the approach to the point affects the evaluation of the derivative.
  • There is a reiteration of the definition of differentiability, emphasizing that all paths approaching (0,0) must yield the same derivative for a function to be considered differentiable at that point.

Areas of Agreement / Disagreement

Participants express differing views on the initial question regarding the existence of such a function g, with some asserting that the question is nonsensical based on definitions, while others provide examples and reasoning that suggest a more nuanced exploration is warranted. The discussion remains unresolved regarding the implications of these definitions.

Contextual Notes

Limitations in the discussion include potential misunderstandings of the definitions of C1 and differentiability, as well as the specific conditions under which the function f(x,y) is evaluated. The discussion does not resolve these ambiguities.

squenshl
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Why is it not possible to find a function g such that g [tex]\in[/tex] C1 at (0,0) and g is not differentiable at (0,0)
 
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Also given that f(x,y) = x|y| how do I show that f [tex]\notin[/tex] C1 at (0,0)
 


squenshl said:
Why is it not possible to find a function g such that g [tex]\in[/tex] C1 at (0,0) and g is not differentiable at (0,0)

As far as I know, a function is defined to be C1 if it is continuously differentiable, so the question doesn't make sense.
 


squenshl said:
Also given that f(x,y) = x|y| how do I show that f [tex]\notin[/tex] C1 at (0,0)

Remember the definition of differentiability: all approaches to the point (0,0) must yield the same derivative. Consider approaching the derivative along the x-axis and then the derivative along the y-axis. Are they equal?
 

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