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Points of Differentiability

  1. Feb 28, 2008 #1
    1. The problem statement, all variables and given/known data
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    2. Relevant equations
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    3. The attempt at a solution

    I think to determine where it's differentiable it has something to do with partial derivatives. But I am just so completely clueless on how to even start this guy off that any tips or minor suggestions on where to even begin would be great. Thanks!
     
  2. jcsd
  3. Feb 28, 2008 #2

    Dick

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    Start with the first one, f. f is just a sum of absolute values. Where is |x| differentiable for x a real variable??
     
  4. Feb 28, 2008 #3
    Everywhere except at x=0.
     
  5. Feb 28, 2008 #4

    Dick

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    Ok, so if ALL of the x_i's are non-zero, then f is differentiable, right? Where is it not differentiable?
     
  6. Feb 28, 2008 #5
    When they're ALL zero? Because if one x_i is non-zero, but the rest are, it's still differentiable?

    I don't know why or how to prove it though, and that's what I need help with.
     
  7. Feb 28, 2008 #6

    Dick

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    If any x_i is equal to zero then the derivative doesn't exist. The ith partial derivative doesn't exist. g is harder, I'm still trying to figure out a good way to describe the set. Try warming up with just two variables. g(x,y)=max(|x|,|y|).
     
  8. Feb 28, 2008 #7
    Alright, I think I understand what's going on with f.

    So then with g, couldn't I try a similar idea since it is also absolute value? Or no?
     
  9. Feb 28, 2008 #8

    Dick

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    It's similar but not the same. Like I say, think about g(x,y)=max(|x|,|y|). That IS NOT differentiable at x=3, y=3. It IS differentiable at x=3, y=0. Once you've figured out why, try and figure out a way to describe all of the points where it is not differentiable.
     
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