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Let $a:=\inf Q\in \mathbb {R}$. There is a bounded linear functional $f$ on $E$ such that for any $x\in E$ one has: $\liminf _ {s\to a} x(s)\le f(x) \le \limsup _ {s\to a} x(s) $. This functional is positive, still vanishes on some functions which are strictly positive on $Q$.