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 $x$. 44. is upper semicontinuous on $\mathbb{R}$ if $∀𝛼∈\mathbb{R}$, the set $\{𝑥∈𝑋:𝑓(𝑥)<𝛼\}$ is open in $X$.