Semi-continuity functions
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 $f$ is upper (respectively, lower) semicontinuous at a point $x_0$ if, roughly speaking, the function values for arguments near $x_0$ are not much higher (respectively, lower) than $f\left(x_0\right)$.

上半连续和下半连续，定义和图很直观，在上面（大的地方）连续，还是在下面（小的地方）连续。

Outer Semicontinuity and Inner semicontinuity

https://mathematix.wordpress.com/2016/06/28/inner-and-outer-semicontinuity-of-multifunctions/