I have some questions while learning about partial derivatives, I didn't found answers neither in books I have nor in the internet. Thanks in advance for your help.
The well-known theorem states that if $f:R^2\to R$ has continuous $\frac{\partial^2f}{\partial x\partial y}$ and $\frac{\partial^2f}{\partial y\partial x}$, then $\frac{\partial^2f}{\partial x\partial y}=\frac{\partial^2f}{\partial y\partial x}$.
But is there any connection between $\frac{\partial^2f}{\partial x^2}$ and $\frac{\partial^2f}{\partial y^2}$ and the mixed derivatives?
I mean does any of basic informations (existence, continuity) about $\frac{\partial^2f}{\partial x^2}$ and $\frac{\partial^2f}{\partial y^2}$ imply the same about mixed derivatives? So the questions are:
does the existence of $\frac{\partial^2f}{\partial x^2}$ and $\frac{\partial^2f}{\partial y^2}$ imply the existence of $\frac{\partial^2f}{\partial x\partial y}$ and $\frac{\partial^2f}{\partial y\partial x}$ ?
does the continuity of $\frac{\partial^2f}{\partial x^2}$ and $\frac{\partial^2f}{\partial y^2}$ imply the continuity of $\frac{\partial^2f}{\partial x\partial y}$ and $\frac{\partial^2f}{\partial y\partial x}$ ?