Let $f:\mathbb{R}\to\mathbb{R}$. $$(\operatorname{cl}f)(x):=\liminf_{x'\to x}f(x')$$ Prove that $\operatorname{cl}(\operatorname{epi}(f))\subset\operatorname{epi}(\operatorname{cl}f)$.
$\forall x,\operatorname{cl}f(x)\leq f(x)$, so $\operatorname{epi}f\subset \operatorname{epi}(\operatorname{cl}f)$. We only need to show that $\operatorname{epi}(\operatorname{cl}f)$ is closed, or equivalently $\operatorname{cl}f$ is lower semicontinuous. But I am stuck here.