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If a real function $f\colon[a,b]\to\mathbb{R}$ is differentiable and its derivative $f'$ is zero, then $f$ is constant. Does this result still hold when $f$ has a weak derivative?

Explicitly, suppose $f\colon[a,b]\to\mathbb{R}$ is an integrable function such that its distributional derivative $Df$ is zero. Does this mean that $f$ is constant?

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    Have you tried invoking the definition of distributional derivative?2012-06-26
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    See also: http://math.stackexchange.com/q/147748 and http://math.stackexchange.com/q/28018/2012-06-26
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    Also related: http://math.stackexchange.com/questions/118525/a-sufficient-condition-for-a-function-to-be-of-class-c2-in-the-weak-sense/120894#1208942012-06-26

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