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Concept
Semi-continuity
Summary
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function is upper (respectively, lower) semicontinuous at a point if, roughly speaking, the function values for arguments near are not much higher (respectively, lower) than
A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point to for some , then the result is upper semicontinuous; if we decrease its value to then the result is lower semicontinuous.
The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899.
Assume throughout that is a topological space and is a function with values in the extended real numbers .
A function is called upper semicontinuous at a point if for every real there exists a neighborhood of such that for all .
Equivalently, is upper semicontinuous at if and only if
where lim sup is the limit superior of the function at the point .
A function is called upper semicontinuous if it satisfies any of the following equivalent conditions:
(1) The function is upper semicontinuous at every point of its domain.
(2) All sets with are open in , where .
(3) All superlevel sets with are closed in .
(4) The hypograph is closed in .
(5) The function is continuous when the codomain is given the left order topology. This is just a restatement of condition (2) since the left order topology is generated by all the intervals .
A function is called lower semicontinuous at a point if for every real there exists a neighborhood of such that for all .
Equivalently, is lower semicontinuous at if and only if
where is the limit inferior of the function at point .
A function is called lower semicontinuous if it satisfies any of the following equivalent conditions:
(1) The function is lower semicontinuous at every point of its domain.
(2) All sets with are open in , where .
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