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 $(0,...,0) \in \mathbb{R}^n$)

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

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