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.

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