Suppose $L \subset A^*$ is context free and $A^*\setminus L$ is also context free. Does it mean, that $L$ is deterministic context free?
If it is not, I would like to see a counterexample (I failed to construct one myself).
Note that the converse is true. Moreover, a complement to a deterministic context free language is also deterministic context free as one can simply change labels on the corresponding deterministic pushdown automaton.