Existence of $f_{xy}$ and $f_{yx}$

Question

I know that if $f:R^2\to R$ has continuous second derivatives, then $f_{xy}=f_{yx}$. But is it possible that $f_{xy}$ is continuous but $f_{yx}$ is not, or that $f_{xy}$ exists but $f_{yx}$ not?

If so, when I calculate $f_{xy}$, I don't know whether $f_{yz}$ exists and continuous, and I cannot use the mixed derivatives theorem, then what is the meaning of this theorem?

Answer 1

Straight from Baby Rudin (theorem 9.41, page 235-6), the theorem is thus:

Theorem: Suppose $f:E\to \mathbb{R}^2$, for $E$ open, and suppose $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$ exist on $E$ and $\dfrac{\partial^2 f}{\partial y\partial x}$ is continuous at some $(a,b)\in E$. Then $\dfrac{\partial^2 f}{\partial x\partial y}$ exists at $(a,b)$ and they are equal.