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Possible Duplicate:
Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$

Let $f : [a,b] \to \mathbb Q$ be a continuous function. Prove that $f$ is a constant function.

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    This is your second question in quick succession. What have you tried?2012-05-13
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    I had some suggestions for your post, but it would have been a repeat of [what Prof Magidin has already suggested](http://math.stackexchange.com/questions/144461/prove-that-there-exists-a-point-a-in-a-such-that-c-a-inf-c-x-x#comment332847_144461). You already have a helpful answer below, but try to keep his advice in mind. Cheers,2012-05-13
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    Although I answered, I downvoted. I sometimes downvote questions that don't show that the poster has tried anything or shown any effort. But if you edit your question, I would be willing to undo that.2012-05-13
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    Also see here: http://math.stackexchange.com/questions/141768/totally-disconnected-space/141771#1417712012-05-13
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    how can you prove that function is constant,even by known fact that it is continuous?Q means rational numbers(represented by ratio form)you need show additional constraints2012-05-13
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    @dato: Note that the continuous image of a connected space is connected. Only connected components of $\mathbb Q$ are rationals. You can prove this without resorting to additional constraints. I do agree, however, that such question is hard to answer if the OP does not supply a survey of their current knowledge.2012-05-13
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    ok thanks @ Asaf Karagila2012-05-13

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