Characterization of the epigraph of a lower semi continuous fuction The goal is to prove that if epigraph of a function $f:X \rightarrow \mathbb{R}$ is closed then it is lower semicontinuous. The epigraph of $f$, $\operatorname{epi} f$ is given as $$ \operatorname{epi} f = \\{ (x, r): f(x) \leq r \\} $$ while a lower semicontinuous function is defined as a function for which: $$ f(\bar{x}) \leq \liminf_{u \rightarrow \bar{x}} f(u) $$ for all $\bar{x} \in X$, where $X$ is a normed space. I have been able to prove the converse but need some pointers on how to proceed in this direction.

Take a sequence $\left \\{ x_{n} \right \\}$ such that $x_{n }\to x_0\in X$ and suppose $\liminf f(x_n)