Suppose that $(X,d)$ be a complete metric space, $f:X \to \mathbb{R}$ is a lower semi-continuous function. Let $\delta: X \to \mathbb{R}$ be a function with the property: $f(x)\leq f(y) \Rightarrow \delta(y)\leq \delta(x), \forall x,y \in X.$ Can we say $\delta$ is upper semi-continuous?