I have a (separable) Banach space $E$ and two closed disjoint sets $F$, $G$ in $E$. Now I wish to prove the existence of a $C^2$-function (Fréchet differentiable) $f:E \to \mathbf R$ that is $1$ on $F$ and $0$ on $G$.
Does someone have a reference for this (if it is possible)? If it is not possible, are there additional conditions on the Banach space to make this possible?