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If $f: U \mapsto \mathbb{R}^n$, $U$ open and connected with $(Df)_p = 0$ for all $p \in U$, then $f$ is clearly constant.

But, with the same assumptions on $U$, what if instead $(D^2f)_p = 0$ (for all $p$.)? Then the $C^1$ Mean-Value Theorem would imply that $$f(q)-f(p)=T(q-p),$$ for all $p,q$. Is there anything more that can be said about $f$? How would one generalize to higher derivatives?

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If the second derivative vanishes everywhere, $f(x)=f_0 + Tx$ for a linear map $T$, so $f$ is affine linear. More generally, if the $n$-th derivative vanishes everywhere, $f$ is a polynomial of degree $n-1$.

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