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[calculus] Continuity of partial derivatives

  1. Sep 8, 2005 #1
    If I am given a function of several variables and a parameter. Such as:
    [tex]f(x,y,z)=\frac{x y z^2}{(x^2+y^2+z^2)^k}[/tex]
    This function is defined to be 0 where it is incontinuous (in [tex](0,0,0)[/tex]).

    How can I conclude for which values of k the function has three continuous partial derivatives?
    I know how to conclude differentiability of the function, but differentiability means partial derivatives exist, not necessarily continuous.

    Thank you.
    Last edited: Sep 8, 2005
  2. jcsd
  3. Sep 8, 2005 #2


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    Differentiability implies continuity, but not the other way arround.
    Differentiability is a stronger condition than continuity and existing partial derivatives is a necessary though not sufficient condition for differentiability.
    For differentiability, you need continuity and existing + continuous partial derivatives.
  4. Sep 8, 2005 #3
    Yes, but as I said I already now how to find differentiability, or for which values of k the function is differentiable.

    It is differentiable for [tex]-\infty<k<\frac{3}{2}[/tex]. Now I want to know for which values of k the partial derivatives are continuous (not the function itself).
  5. Sep 8, 2005 #4


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    First, find the partial derivatives. The, apply the definition of continuity to the three functions (each partial derivative).
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