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 subset $K$ of $\mathbb R$ of cardinality continuum, the set of those $n$ such that $f_n(K)$ is not the whole of $\mathbb R$ is finite?