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 ${\displaystyle x}$ is near ${\displaystyle x_{0}}$ then continuity says "${\displaystyle f(x)}$ is near ${\displaystyle f(x_{0})}$." Upper semicontinuity relaxes the condition to "${\displaystyle f(x)}$ is near or below ${\displaystyle f(x_{0})}$." Lower semicontinuity relaxes it to "${\displaystyle f(x)}$ is near or above ${\displaystyle f(x_{0})}$".

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

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

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

Add pictures.

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