For $C[0,1]$ the space of all continuous functions $f:[0,1]\to\mathbb{R}$, Show that, in $C[0,1]$, the functions with $f(\mathbb{Q})\subseteq \mathbb{Q}$ are dense. I'm having trouble understanding what dense means in relation to functions, so am not sure how to answer this! Any help would be appreciated :)
I am trying to prove the following:  In $C[0,1]$ the functions that preserve the rationals (i.e. $f(\mathbb{Q})\subseteq \mathbb{Q}$) are dense.  So far, I have not made much progress- I am aware that $A$ is dense in $B$ iff $A$ has non-empty intersection with each open set $U$ in $B$ but am not sure how to apply this definition. I have also thought about showing that the closure of $X$ (where $X$ is the set of functions that preserve the rationals) is equal to $C[0,1]$ by a double-inclusion proof but I have also made no progress with this approach. Am I on the right track or should I change my approach?
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