[calculus] Continuity of partial derivatives

[Ahmes]

Hello,
If I am given a function of several variables and a parameter. Such as:
$$f(x,y,z)=\frac{x y z^2}{(x^2+y^2+z^2)^k}$$
This function is defined to be 0 where it is incontinuous (in $$(0,0,0)$$).

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.

 

[TD]

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.

 

[Ahmes]

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 $$-\infty<k<\frac{3}{2}$$. Now I want to know for which values of k the partial derivatives are continuous (not the function itself).

 

[TD]

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