It's a question I've been thinking about but I can't find an easy answer. I think it will be interesting. Can there be a countable collection of real valued functions $f_1, f_2 , ... $ such that for any uncountable subset $K$ of $\mathbb R$ the set of those $n$ such that $f_n(K)$ is not the whole of $\mathbb R$ is finite? Clearly, this can't be satisfied by finitely many functions, and easily if we allow the collection to be uncountable.

Thanks!