Suppose $f:V \subset R^n \to R^m$ is a smooth function such that $D_f = A$ for some constant matrix $A$ in $V$, and $V$ is open. My question is, is it true that $f(x) = y + Ax$ for some constant $y \in R^m$? If so, why?
2026-03-27 19:31:58.1774639918
If the differential of a function is constant, does the function has to be affine?
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This is not true, but it’s almost true. The function $$g(x)=f(x)-Ax$$ has total derivative equal to $0$ everywhere, so it must be constant on every connected component of $V$. If $V$ is connected, the result thus follows.