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Silly question. Suppose that $f:\mathbb{R}^n\to\mathbb{R}^n$, and $T$ is a linear transformation such that $f(x+h)-f(x)-T(h)=0$ for all $x,h$. Does it follow that $f$ is linear?

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If $v_0\in \Bbb R^n$ and $f$ satisfies the condition so does $f-v_0$, so $f$ is not necessarily linear. But taking $x=0$, we get $f(h)=T(h)+f(0)$, so we can write $f=T+v_0$, where $v_0$ is a fixed vector and $T$ linear. It's an affine map.