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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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doit
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    This is your second question in quick succession. What have you tried? – Brian M. Scott May 13 '12 at 06:23
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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. You already have a helpful answer below, but try to keep his advice in mind. Cheers, – Dylan Moreland May 13 '12 at 06:23
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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. – davidlowryduda May 13 '12 at 06:25
  • Also see here: http://math.stackexchange.com/questions/141768/totally-disconnected-space/141771#141771 – Asaf Karagila May 13 '12 at 06:26
  • 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 constraints – dato datuashvili May 13 '12 at 06:29
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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. – Asaf Karagila May 13 '12 at 06:33

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HINT:

Is $[a,b]$ connected? Is $\mathbb{Q}$?

davidlowryduda
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