# Is union of context-free and non-context-free languages non-context-free?

It is known that union of two context-free languages is also context-free.

But how about union of context-free and non-context-free languages?

Not in general. Consider a language $L$ over some alphabet $\Sigma$ such that neither $L$ nor its complement $\overline{L}$ is a decidable language. It is obvious that neither of these languages can be context-free. But $L \cup \overline{L} = \Sigma^*$, which is a context-free language.
And consider $\emptyset$, which is trivially a context-free language. Clearly $L \cup \emptyset = L$ is not a context-free language.