Show that, in $C[0,1]$, the functions with $f(\Bbb Q) \subseteq \Bbb Q$ are dense.

Question

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 :)

Answer 1

A subset $A\subseteq C[0,1]$ of functions is called dense in $C[0,1]$ if $$ \forall \varepsilon>0 \,\forall g\in C[0,1]\, \exists h \in A: \sup_{x\in [0,1]} |h(x)-g(x)|< \varepsilon. $$ In your case, the set $A$ would be $$ A=\{f\in C[0,1] | f(\Bbb Q\cap [0,1]) \subseteq \Bbb Q \}. $$