For an antisymmetric function $f:\mathbb R\rightarrow \mathbb R$ (i.e. f(x)=-f(-x)) we have: necessary condition for the differential of f of order $r$ to not vanish at $0$ is that $r$ is odd.
My question is: what if I consider an antisymmetric $f:\mathbb R^n\rightarrow \mathbb R$ with $n\geq 2$ ?
Here antisymmetric means: $f(x)= sign(\pi) f(\pi x)$ for every signed permutation $\pi$ of the coordinates ( a signed pemutation is a permutation which is also allowed to change the sign of the coordinate, I hope it's clear what I mean, i don't know if it's standard terminology).
I think that now we have: necessary condition for the differential of f order $r$ to not vanish at $0$ is that $r=k n$ with $k$ odd.
Is this true? In case, how to prove it? Any help would be appreciated.