# Mixed partial derivatives-a counterexample

The well known theorem of Schwarz states that if $f \in C^k(U)$ then all partial derivatives up to $k$-th order of $f$ are equal. There is a well known example of function of class $C^2(R^2)$ such that $\frac{\partial^2f}{\partial x\partial y}(0,0) \neq \frac{\partial^2f}{\partial y\partial x}(0,0)$. I would like to know a most general example: where $f$ is of $n$-variables, have all derivatives up to $k$-th order and all mixed derivatives of $k$-th order are different for a given point in $\mathbb{R}^n$ (for example for $(0,...,0) \in \mathbb{R}^n$)

• I think you want mixed partial derivatives in there – qbert May 10 '16 at 17:02
• What do you mean will all derivatives being different? – Git Gud May 10 '16 at 17:23
• Can you integrate the well-known example $k-2$ times? – Matthew Leingang May 10 '16 at 17:33
• My example is for $2$ variables and moreover I would like to have some visible explicit formula for this function – truebaran May 10 '16 at 18:46