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Order of partial derivatives

  1. Oct 4, 2004 #1
    Given a scalar-valued function [tex]f=f(x,y)[/tex], if it's true that [tex]\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial x}[/tex], what does that tell about function [tex]f[/tex]? Does it mean that it's continuous, or does it need to be smooth, or...?
  2. jcsd
  3. Oct 4, 2004 #2
    I'm presuming that the correct answer is that the function [tex]f[/tex] must be continuous.
  4. Oct 4, 2004 #3
    You are right. The partial derivatives upto that order should exist and be continuous at the point under consideration.
  5. Oct 4, 2004 #4
    Its more than continous, actually it tells you that the function f is differentiable.

    [tex]f(x,y)\in C^{1}[/tex]
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