relations between the derivatives of second rank

Question

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:

1. 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 $\frac{\partial^2f}{\partial y\partial x}$ ?

2. 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 $\frac{\partial^2f}{\partial y\partial x}$ ?

Answer 1

The answer to 1. is no: Define $f(x,y) = y|x|.$ Then $f$ vanishes on the $x$ and $y$ axes, hence $f_{xx}(0,0) = 0 = f_{yy}(0,0).$ But note $f_y(x,0) = |x|.$ Thus $f_{yx}(0,0)$ fails to exist.