I know that $\frac {\partial^2 f}{\partial x \partial y} = \frac {\partial^2 f}{\partial y \partial x}$ when $\frac {\partial f}{\partial x}$, $\frac {\partial f}{\partial y}$ exist and are continuous, via the Schwarz's theorem.
I wonder if this logic is valid for higher order derivatives or not, such as:
$\frac {\partial^3 f}{\partial x \partial y \partial z} = \frac {\partial^3 f}{\partial x \partial z \partial y} = \frac {\partial^3 f}{\partial y \partial x \partial y} = \frac {\partial^3 f}{\partial y \partial z \partial y} = \frac {\partial^3 f}{\partial z \partial x \partial y} = \frac {\partial^3 f}{\partial z \partial y \partial x}$