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User:Editeur24/uppersemicontinuity

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Needs better graphs. Have two points of discontinuity, so it isn't left- and right-

https://math.stackexchange.com/questions/1182795/what-is-the-intuition-for-semi-continuous-functions has a nice discussion of intuition. https://planetmath.org/Semicontinuous1

In mathematical analysis, semi-continuity (or semicontinuity) is a property of functions that is weaker than continuity. If is near then continuity says " is near ." Upper semicontinuity relaxes the condition to " is near or below ." Lower semicontinuity relaxes it to " is near or above ".


Upper semicontinuity at a point

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Note that X is a topological space, and need not be a metric space.



Characterizations

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FOOTNOTE One might think that the open-set definition fails when the domain X has boundary points, e.g. X = {x \geq 0 \in \R} because the set of x's with f(x) <y could be only half-open. This is false, however, because the standard definition of a domain is as the intersection of ... ask Chris.

Another equivalent approach to definition that is only applicable to metric spaces, not topological spaces generally (because it will use the distance $\epsilon$, undefined without a metric) goes as follows. from topological space to the extended real numbers (that is, including is continuous at point if and only if for any given there is some neighbourhood of such that for all we have .

The function is upper semi-continuous at if and only if for any given there is some neighbourhood of such that for all we have .

The function is said to be lower semi-continuous at if and only if for any given there is some neighbourhood of such that for all we have .

Add pictures.


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