For a lower semi-continuous function $f: \mathbb R^n\to \mathbb R$, show $\{z\in \mathbb R^n : f(z)> r\}$ is open for $r \in \mathbb R$

Question

Let $f:\mathbb R^n\to \mathbb R$ be a lower semi-continuous function, how to show for any constant $r$ $U:=\{z\in \mathbb R^n : f(z)> r\}$ is open ?

Answer 1

Consider a sequence $x_n \in \mathbb{R}^n\backslash U = E$ and assume $x_n\rightarrow x$. Then $f(x) \le \liminf f(x_n) \le r$, so $x\in E$ and $E$ is closed.