Definition (Semicontinuous)

Let $X$ be a topological space and let $f$ be a function from $X$ into $\mathbb{R}$.

1. $f$ is lower semicontinuous on $\mathbb{R}$ if:
2. $∀𝛼∈\mathbb{R}$, the set $\{𝑥∈𝑋:𝑓(𝑥)>𝛼\}$ is open in $X$ or;
3. $\forall x\in X$ $$\liminf_{ n \to \infty } f(x_{n})\ge f(x)$$for any sequence $\{ x_{n} \}$ in $X$ that converges to $x$.
4. is upper semicontinuous on $\mathbb{R}$ if $∀𝛼∈\mathbb{R}$, the set $\{𝑥∈𝑋:𝑓(𝑥)<𝛼\}$ is open in $X$.