The indicator function of any open set is lower semicontinuous.
2.
One can differentiate many continuity concepts, primarily closed graph property and upper and lower semicontinuous real-valued function ).
3.
Nevertheless, lower semicontinuous multifunctions usually possess continuous selections as stated in the Michael selection theorem, which provides another characterisation of paracompact spaces.
4.
Every positive form ? extends uniquely to the linear span of non-negative bounded lower semicontinuous functions " g " by the formula
5.
Let X be a normal topological space and let g, h \ colon X \ to \ mathbb { R } be functions with g upper semicontinuous, h lower semicontinuous and g \ leq h.
6.
"A sequentially weakly lower semicontinuous coercive functional on a reflexive Banach space has a minimizer . " A slight generalization appears in Giusti ( 2003 ), " Direct methods in the calculus of variations ", World Scientific.
7.
However, in convex analysis, the term " indicator function " often refers to the characteristic function, and the characteristic function of any " closed " set is lower semicontinuous, and the characteristic function of any " open " set is upper semicontinuous.
8.
Using lower and upper Hausdorff uniformity we can also define the so-called "'upper "'and "'lower semicontinuous maps in the sense of Hausdorff "'( also known as "'metrically lower / upper semicontinuous maps "').
9.
If X = \ mathbb R ^ n is a Euclidean space ( or more generally, a metric space ) and \ Gamma = C ( [ 0, 1 ], X ) is the space of curves in X ( with the length L ( \ alpha ), is lower semicontinuous.
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