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Can there be a continuous non-constant function with only rational values defined on an interval in R? Or is the whole property meaningless?

user75907
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    Any continuous image of a connected space is connected, the only connected subspaces of $\mathbb{R}$ are a point and intervals (possibly infinite). – Yuchen Liu May 04 '13 at 10:00
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    Continuous functions on the reals have the intermediate value property; if they take on the values $a$ and $b$ then they take on every value between $a$ and $b$. – Gerry Myerson May 04 '13 at 10:02

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No. Assume there was such an $\,f \,:\, [a,b] \to \mathbb{Q}$. If its' non-constant there are $x,y \in [a,b]$ with $f(x) \neq f(y)$. You can then pick $c$ from $[f(x),f(y)] \setminus \mathbb{Q}$, since that set is non-empty. Now, can you show that there's an $z$ between $x$ and $y$ for which $f(z)=c$?. If so, you have a contradiction, since $f(z) = c \notin \mathbb{Q}$ per the choice of $c$.

Hint: Use that $f$ is continous for the missing part.

fgp
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