let $E$ be a topological space and $\phi:E\to\mathbb{R}\cup\{\infty\}$. then the following are equivalent:
1) $\liminf_{x\to x_0}\phi(x)\geq\phi(x_0)$
2) $epi(\phi):=\{(x,\lambda)\in E\times\mathbb{R}:\phi(x)\leq\lambda\}$ is closed in $E\times\mathbb{R}$
3) $C_{\lambda}:=\{x\in E:\phi(x)\leq\lambda\}$ is closed in $E$ $\forall\lambda\in\mathbb{R}$
have fun!