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Sufficient condition for differentiability of a function of two variables

  1. Oct 4, 2009 #1
    Is there a convenient sufficient condition for knowing whether a function of two variables is differentiable? Isn't it something like if both the partial derivatives exist and are continuous, you know the derivative [itex]\mathbf{D}f[/itex] exists?
     
  2. jcsd
  3. Oct 5, 2009 #2

    HallsofIvy

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    Yes, that is correct. A function is differentiable at a point if and only if the partial derivatives exist and are continuous in some neighborhood of the point.
     
  4. Oct 5, 2009 #3

    quasar987

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    I'm not sure about the "only if" part.

    Take for instance f(x) = x² if x is rational and =-x² otherwise. Then f is differentiable at 0 but nowhere else.
     
  5. Oct 5, 2009 #4
    the derivative of a function can be discontinuous
     
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