Let $X = C([-1,1])$ with the usual norm $\lVert f \rVert_{\infty} = \sup_{t \in [-1,1]} |f(t)|$. Suppose $N = \{ f \in X : f \, \, \text{is neither even nor odd} \}$. Show that $N$ is not meager.
I've shown that the set of even functions and odd functions on $X$ is meager. So I think that the set of neither even nor odd functions should not be meager. I'm trying to show this by showing that $N$ is closed and showing that the closure of $N$ is not empty. I'm not making any progress so I don't know if this is correct.