Semidiscontinuous

Semidiscontinuous is a nonstandard or older term that appears in some texts to refer to semicontinuity, a fundamental notion in real analysis and topology. In modern usage, the standard terms are upper semicontinuous and lower semicontinuous.

Definitions and basic ideas

A function f defined on a topological space X with values in the extended real line is upper semicontinuous (USC) at a point x0 if the limit superior of f as x approaches x0 does not exceed f(x0). Equivalently, for every ε > 0 there exists a neighborhood of x0 such that f(x) < f(x0) + ε for all x in that neighborhood. A function is upper semicontinuous on a set if it is USC at every point of the set. A function is lower semicontinuous (LSC) at x0 if the limit inferior as x→x0 is at least f(x0); equivalently, for every ε > 0 there exists a neighborhood where f(x) > f(x0) − ε for all x in that neighborhood. If a function is USC at every point, it is USC; if it is LSC at every point, it is LSC. A function that is both USC and LSC is continuous.

Relation to the term semidiscontinuous

Some authors use semidiscontinuous to mean semicontinuous in general, or to indicate semicontinuity without specifying whether it is upper or lower. In many contexts, semicontinuity plays a key role in existence results for optimization, variational problems, and functional analysis. The semicontinuous envelope concepts—the greatest USC minorant and the least LSC majorant—are often employed to study approximations and convergence.

Examples

The function f(x) = 0 for x ≠ 0 and f(0) = 1 is upper semicontinuous but not lower semicontinuous. Conversely, g(x) = 0 for x ≠ 0 and g(0) = −1 is lower semicontinuous but not upper semicontinuous. A continuous function is both USC and LSC.