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[calculus] Continuity of partial derivatives

  • Thread starter Ahmes
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  • #1
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Hello,
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.
 
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Answers and Replies

  • #2
TD
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Ahmes said:
I know how to conclude differentiability of the function, but differentiability means partial derivatives exist, not necessarily continuous.
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.
 
  • #3
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TD said:
For differentiability, you need continuity and existing + continuous partial derivatives.
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).
 
  • #4
TD
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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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