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How does one show that the function is differentiable?

  1. Jan 1, 2014 #1
    I'd like to show that if [itex]\alpha>\frac{1}{2}[/itex] then [itex](x^2+y^2)^\alpha[/itex] is differentiable at [itex](0,0)[/itex].

    The usual way is to show that the partial derivatives are continuous at [itex](0,0)[/itex].

    Yet I am a little confused how to show that [itex]2x\alpha(x^2+y^2)^{\alpha-1}[/itex] is continuous at [itex](0,0)[/itex]. I have tried working it out by definition, yet it seems to be a mess.

    Any hints are very appreciated!
     
  2. jcsd
  3. Jan 1, 2014 #2
    I might be wrong, so you should wait until others give advice, but I believe something that can get you started would be to first break this into two cases for the values of ##\alpha##. After that, I would use a squeeze theorem argument for one of the cases.
     
  4. Jan 1, 2014 #3

    Simon Bridge

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  5. Jan 1, 2014 #4
    Simon Bridge,
    thank you, it's all clear now!
     
  6. Jan 2, 2014 #5

    Simon Bridge

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    No worries - happy New Year.
     
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