# An extension for lower semi continuous lower bounded real valued functions class

Let $$(X,d)$$ be a complete metric space. I need some explanations about the class of all functions like $$f$$ which have $$f:X \to \mathbb{R}\cup\{ +\infty\}$$, be a lower bounded and, for all $$y \in X$$ we have the set $$\{x \in X : f(x) \leq f(y)\}$$ is closed.

I know this class is a Pure super-set of family of lower semi continuous lower bounded functions, and the subset of (sequentially) lower monotone lower bounded functions. I am not sure, if this class is a pure subset of (sequentially) lower monotone lower bounded or not.

I think it is better to mention my meaning by sequentially lower monotonicity : f is said to be (sequentially) lower monotone(lower semi continuous from above), if for all decreasing sequence $$\{ f(x_n)\}_{n \in \mathbb{N} }$$, the condition $$\lim_{n\to \infty} x_n=x_0$$ implies $$f(x_0)\leq \liminf_{n \to \infty}f(x_n).$$ Some references called this class to partially lower semi continuous, and some others called this lower semi continuous from above.

• If I understand you correctly, the function $f:\mathbb{R}\to\mathbb{R}, x\mapsto x$ is not in your class (because it is not lower bounded). However, $f$ would be lower semi continuous. Wouldn't this be a contradiction to your statement, that your class is a superset of the family of lower semi continuous functions? Oct 25, 2018 at 15:17
• Thats right. I should correct my meaning. I meant, my class is superclass for the class of continuous lower bounded function. Oct 25, 2018 at 16:49

$$f(x)= \left\{ \begin{array}{ll} e^{-x}+\frac{1}{2} & x\geq 0 \\ e^x & x<0 \\ \end{array} \right .$$ It is clear that $$f$$ is lower monotone, and $$\{x \in \mathbb{R}: f(x)\leq f(\ln(2))\}$$ is not closed. If we redefine $$f(0):=\frac{5}{4}$$, the results will hold, but this f is not upper semi continuous, even it is not upper monotone.