Let $f: [0,1]\to \mathbb{R}$ be a lower semi-continuous function, then
$$ \liminf_{x\to a} f(x) \geq f(a), \forall a \in [0,1]$$
I have to prove that $f$ attains its minimum on $[0,1]$, that is:
$\exists x_0 \in [0,1]$ such that $f(x_0) \le f(x)$, $\forall x \in [0,1]$.
This is a problem from a past qualifying exam in Measure Theory. I'm trying to solve it but I do not know how I should begin with this problem. I did not understand where the Inf is taken. What means that limit?