I would like to construct some counterexamples:
$E$ is an infinite dimensional normed space. Let $C, D$ be nonempty convex subsets in $E$ such that $ C\cap D=\emptyset. $ There is no vector $f\in E^*\setminus \{0\}$ and $\alpha\in \mathbb{R}$ such that $ f(x)\leq \alpha\leq f(y) \quad \forall x\in C, \quad \forall y\in D. $
$E$ is an infinite dimensional normed space. Let $C, D$ be nonempty convex subsets in $E$ such that $ C\cap D=\emptyset $ and $C$ is compact. There is no vector $f\in E^*\setminus\{0\}$ and $\alpha\in \mathbb{R}$ such that $ f(x)< \alpha< f(y) \quad\forall x\in C, \quad \forall y\in D. $
Thank you for all comments and helping.