What is lower semicontinuous?

A function is said to be lower semicontinuous if, intuitively, its graph does not have "jumps." More formally, a function f: X -> R from a topological space X to the real numbers is lower semicontinuous if, for every real number a, the set of points {x in X | f(x) > a} is an open set.

Lower semicontinuity is a useful concept in optimization, functional analysis, and other areas of mathematics. It is a generalization of the concept of continuity and allows for functions that may have discontinuities, but not "jumps" that are too extreme.

Properties of lower semicontinuous functions include: they are always measurable, convex, and continuous from below. They can also be characterized in terms of their epigraph, the set of points lying above the graph of the function.

Overall, lower semicontinuous functions provide a flexible tool for analyzing and understanding functions that may not be perfectly smooth or continuous.