Upper-semicontinuous function

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A function $u:X\to[-\infty,\infty$ ) on a topological space is upper-semicontinuous if

\limsup_{y\to x}u(y)\le u(x)

for every $x\in X$ . Equivalently, every strict sublevel set $\{x:u(x)<a\}$ is open.

Sequential form

On a metric space, upper semicontinuity is equivalent to $\limsup_{n\to\infty}u(x_n)\le u(x)$ whenever $x_n\to x$ .

Role in potential theory

The upper-semicontinuity condition prevents upward jumps from being hidden by averaging. Together with a sub-mean inequality, it is part of the definition of subharmonic and plurisubharmonic functions.

References

Thomas Ransford, Potential Theory in the Complex Plane, Cambridge University Press, 1995. DOI record.