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I have been attempting to do the following question by contradiction. However, I just got stuck at where to use the given continuity condition. It would be really appreciated if you can possibly give a further hint to solving this problem.

Here it goes

Let $f(x):\mathbb{R} \rightarrow \mathbb{Q}$ is a continuous function. Prove that $f$ is a constant.

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    Can you find a real number between two given rationals?2012-05-05
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    More to the point, can you find an *irrational* number between two rationals?2012-05-05
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    Yes, it was my initial idea of using Density Theorem. However, I do not see the connection between that with the function.2012-05-05
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    Look at [this](http://math.stackexchange.com/q/55638/8271)2012-05-05

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