Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}.$ $f$ is a constant function?
Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}$
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real-analysis
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0Are you asking if $f$ must be a constant function, or if a constant function satisfies the criterion? – 2012-06-22
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5@jbowman: Undoubtedly the former. – 2012-06-22
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3@BrianM.Scott: One hopes so, but I've been surprised a time or two! – 2012-06-22
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0If $f$ is differentiable on $\mathbb{R}$, then $f'(x)$ exists on $\mathbb{R}$, and if $f'$ is discontinuous at $x=a$, then it must be an *essential* discontinuity. That is, there cannot be a jump discontinuity there. But I'm having trouble making that into a proof that $f$ has to be constant. – 2012-06-22
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1This question is related: http://math.stackexchange.com/questions/151931/set-of-zeroes-of-the-derivative-of-a-pathological-function – 2012-06-22
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0$f'$ maps a set of irrational numbers to an interval. – 2012-06-22