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Continuous partial derivative?

  1. Jun 13, 2010 #1
    My textbook describes how some functions are not well approximated by tangent planes at a particular point. For example

    f(x)= xy / (x^2 + y^2) for x /= 0
    0 for x = 0

    at (0,0) the partial derivatives exist and are zero but they are not continuous at 0. What exactly is a 'continuous partial derivative' in two variables? How do you visualize this?
  2. jcsd
  3. Jun 14, 2010 #2


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  4. Jun 14, 2010 #3
    In the case of that example, is it not differentiable at zero because its not continuous there?
  5. Jun 14, 2010 #4


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    A function is only differentiable at zero if a unique tangent plane can be assigned there.

    Differentiability IMPLIES existence of partial derivatives, but the converse does not hold.
  6. Jun 14, 2010 #5
    Thanks! By the way, nice job on the 9,999 posts :)
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