User:Editeur24/uppersemicontinuity

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  $$x$$ is near $$x_0$$ then  continuity says "$$f(x)$$ is near $$f(x_0)$$." Upper semicontinuity relaxes the condition to "$$f(x)$$ is near or below $$f(x_0)$$." Lower semicontinuity relaxes it to "$$f(x)$$ is near or above $$f(x_0)$$".

Upper semicontinuity at a point
Note that X is a topological space, and need not be a metric space.

Characterizations
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. $$f$$   from  topological space $$X$$ to the extended real numbers (that is, including  $$\{-\infty, \infty\})$$ is continuous at point $$x_0$$  if and only if   for any given $$\epsilon>0$$ there is some neighbourhood $$U$$ of $$x_0$$ such that for all $$x \in U$$ we have  $$f(x_0) - \epsilon < f(x) < f(x_0) + \epsilon$$.

The function $$f $$ is  upper semi-continuous at   $$x_0 $$  if and only if for any given $$\epsilon>0$$ there is some neighbourhood $$U$$ of $$x_0$$ such that for all $$x \in U$$ we have $$ f(x) < f(x_0)+\epsilon$$.

The function $$f $$ is said to be  lower semi-continuous at   $$x_0 $$  if and only if for any given $$\epsilon>0$$ there is some neighbourhood $$U$$ of $$x_0$$ such that for all $$x \in U$$ we have $$f(x_0) - \epsilon < f(x)$$.

Add pictures.

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