What does equal mixed partial derivatives indicate about a function?

[kuba]

Given a scalar-valued function f=f(x,y), if it's true that \frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial x}, what does that tell about function f? Does it mean that it's continuous, or does it need to be smooth, or...?

 

[kuba]

I'm presuming that the correct answer is that the function f must be continuous.

 

[neutrino]

You are right. The partial derivatives upto that order should exist and be continuous at the point under consideration.

 

[ReyChiquito]

Its more than continous, actually it tells you that the function f is differentiable.

f(x,y)\in C^{1}