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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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marked as duplicate by Martin Sleziak, Dylan Moreland, Asaf Karagila, mixedmath, Rahul May 13 '12 at 6:56

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

This is your second question in quick succession. What have you tried? –  Brian M. Scott May 13 '12 at 6:23
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 6:23
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. –  mixedmath May 13 '12 at 6:25
Also see here: math.stackexchange.com/questions/141768/… –  Asaf Karagila May 13 '12 at 6:26
@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 6:33

1 Answer 1


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

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