uppersemicontinuity

Upper semicontinuity is a property of a real-valued function defined on a topological or metric space that governs how the function can jump upward. A function f is upper semicontinuous (USC) at a point x0 if the limit superior of f(x) as x approaches x0 does not exceed f(x0); equivalently, limsup_{x→x0} f(x) ≤ f(x0). A practical formulation is: for every ε > 0 there exists a neighborhood U of x0 such that f(x) ≤ f(x0) + ε for all x in U.

A function is upper semicontinuous on its domain if it is USC at every point. Several equivalent characterizations help intuition and proofs: for every a ∈ R, the set {x | f(x) < a} is open; equivalently the superlevel sets {x | f(x) ≥ a} are closed. In particular, continuous functions are USC, and USC functions may have downward jumps but not upward jumps that violate the limit condition.

USC interacts with lower semicontinuity: f is upper semicontinuous if and only if the negation −f is lower semicontinuous. This parallels the relationship between continuity, upper and lower semicontinuity.

Examples and non-examples illustrate the concept. A classic USC but not continuous function is f(x) = 0 for x < 0 and f(x) = 1 for x ≥ 0 on the real line; it jumps upward at 0 but satisfies limsup_{x→0} f(x) ≤ f(0). Conversely, a function with an upward jump violating the limsup condition would fail to be USC.

Applications of USC include optimization theory, where on a compact domain a USC function attains its maximum (provided it is bounded above). USC also arises in variational problems and in the analysis of convergence of function sequences.