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I know that for a function $f$ all of its partial derivatives are $0$.

Thus, $\frac{\partial f_i}{\partial x_j} = 0$ for any $i = 1, \dots, m$ and any $j = 1, \dots, n$.

Is there any easy way to prove that $f$ is constant? The results seems obvious but I'm having a hard time expressing it in words explicitly why it's true.

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    Think of it this way: you can ‘walk’ from any point of $\Bbb R^n$ to any other point along lines parallel to the coordinate axes, and $f$ is constant along those lines.2012-04-11
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    This assumes of course that the domain of $f$ (which you have not specified) contains the required paths.2012-04-11
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    @ChrisEagle : And in particular, it doesn't contain those paths if it's not connected.2012-04-11

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