Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper lower semi continuous function, which is bounded from below, $Z=\{x \in X: f(x)=\inf f\}$. Let for all $x \in X \setminus Z$, there exists $y\in X\setminus \{x\}$ such that $f(y)+d(x,y)\leq f(x).$ Then $Z\not= \emptyset.$

My question is that if we replace "lower pseudo-continuous" instead of "lower semi continuous" in the above theorem, whether or not the result hold?

$f$ is said to be lower pseudo-continuous on $X$, if for all $y \in X$, the set $\{x \in X : f(x) \leq f(y)\}$ would be a closed subset of $X$.

Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded from below, $Z=\{x \in X: f(x)=\inf f\}$. Let for all $x \in X \setminus Z$, there exists $y\in X\setminus \{x\}$ such that $f(y)+d(x,y)\leq f(x).$ Then $Z\not= \emptyset.$

My question is that if we replace "lower pseudo-continuous" instead of "lower semi continuous" in the above theorem, whether or not the result hold?

$f$ is said to be lower pseudo-continuous on $X$, if for all $y \in X$, the set $\{x \in X : f(x) \leq f(y)\}$ would be a closed subset of $X$.