I am trying to understand one commonly stated necessary and sufficient condition for a function to be (lower) semicontinuous:
$f(x) \leq \liminf_{y \rightarrow x}{f(y)} \iff \forall a \in \mathbb{R} \quad Z(f,a) \equiv \{ x : f(x) \le a \} \mbox{ is closed} $
I can easily prove the forward direction with limits of sequences of elements taken from $Z(f,a)$, but I'm having trouble proving the backward direction. I can prove the following, however,
$f(x) \leq \liminf_{y \rightarrow x}{f(y)} \iff \mbox{epi}(f) \equiv \{(x,y) : f(x) \le y\} \mbox{ is closed} $
I would like to be able to make some connection between $\mbox{epi}(f)$ and $Z(f,a)$, but none of the usual countable union properties of closed sets can help me. Is this the wrong path to be going down, or is there perhaps some simple step I'm missing?