Lower semicontinuous functions: Difference between revisions

Revision as of 20:45, 26 October 2020

Let $X$ be a metric space (or more generally a topological space). A function $f:X\to \mathbb {R} \cup \{+\infty \}$ is lower semicontinuous if

\{x\in X:f(x)>a\}=f^{-1}\left((a,+\infty ]\right)

is open in $X$ for all $a\in \mathbb {R}$ .^[1]

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-Here is a page where we learn about lower semicontinuous functions!!!
+Let <math> X </math> be a metric space (or more generally a topological space). A function <math> f : X \to \mathbb{R} \cup \{ +\infty \} </math> is '''lower semicontinuous''' if
+:<math> \{ x \in X : f(x) > a \} = f^{-1} \left( ( a , +\infty ] \right) </math>
+is open in <math> X </math> for all <math> a \in \mathbb{R} </math>.<ref name="Craig">Craig, Katy. ''MATH 201A HW 3''. UC Santa Barbara, Fall 2020.</ref>
-<math> f(x) = x^2 </math>
+==References==