Let $X$ be a metric space. A function $f \colon X \mapsto \mathbb{R} \cup\lbrace -\infty, \infty \rbrace $ is lower semicontinuous at a point $x_{0}$, if $$ \liminf \limits_{x \rightarrow x_{0}} f(x) \ge f(x_{0}). $$ Similarly, we can define upper semicontinuous functions. But I want to know, what is the motivation for defining these functions. Do, these functions arises in some real life applications or make sense only theoretically.



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