The well known theorem of Schwarz asserts the following: suppose that $f:U \to \mathbb{R}$ where $U \subset \mathbb{R}^n$ is $C^k$ function and pick some sequence $(j_1,...,j_k)$ of length $k$ where each $j_i \in \{1,2,...,n\}$. Then all mixed partial derivatives of $k$-th order over variables $x_{j_i}, i=1,2,...,k$ coincide, no matter in which order we diffrentiate. I would like to see the following example: function $f:\mathbb{R}^n \to \mathbb{R}$ which has all partial derivatives up to $k$-th order (where $k$ is a fixed positive integer) but for every point $x_0 \in \mathbb{R}^n$ all $k$-th order derivatives has different values. In other words I would like to see an example where for every permutation of variables we get different value of partial derivative and this happens not only in one but in every point.
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1I don't think such an example will be possible. If the partial derivatives do not commute at some point then the partial derivatives of $f$ cannot all be continuous at that point, but there are restrictions on the points of discontinuity of a differentiable function. The answer to this question describes what the points of discontinuity can look like for one dimension. For your question you are looking for higher derivatives in higher dimension, it may be that these restrictions remain in this setting. – s.harp Jun 02 '20 at 10:13