Questions tagged [semicontinuous-functions]

Tag for questions about upper-semicontinuous and lower-semicontinuous functions.

Some authors us the name semicontinuity also for multifunctions, many authors use the name hemicontinuous. This tag is specifically for questions about functions, for multifunctions use multivalued-functions.

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$f$ is LSC at $x$ if and only if $\lim_{\delta \to 0}\inf\{f(y) | y \in B(x,\delta)\}=f(x)$

Asuume $f$ is LSC(lower semicontinuous) at $x$ . If $t\lt f(x)$, then there exsits $\delta \gt 0$ such that $t\lt f(y)$ for all $y \in B(x,\delta)$. Thus, $t \le \inf\{f(y)| y\in B(x,\delta)\}$ and we conclude that $t$ $\le$ $\lim_{\delta \to 0}$…

semicontinuous-functions

asked May 14 '18 at 17:35

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