semicontinuity

Semicontinuity is a form of continuity that is weaker than the standard notion, studied for real-valued functions on topological spaces. There are two dual notions: lower semicontinuity and upper semicontinuity. A function is lower semicontinuous (lsc) if it cannot jump downward at a point; a function is upper semicontinuous (usc) if it cannot jump upward.

In metric or topological spaces, a common formulation uses limit behavior. A function f is lsc at a point x0 if the limit inferior of f(x) as x approaches x0 is at least f(x0): liminf_{x→x0} f(x) ≥ f(x0). Equivalently, for every ε > 0 there exists δ > 0 such that d(x, x0) < δ implies f(x) > f(x0) − ε. A function is usc at x0 if limsup_{x→x0} f(x) ≤ f(x0); equivalently, for every ε > 0 there exists δ > 0 such that d(x, x0) < δ implies f(x) < f(x0) + ε. These definitions extend to extended real-valued functions and may be stated in terms of nets or sequences in spaces that are not first-countable.

Continuity implies both lower and upper semicontinuity, and a function that is both lsc and usc is continuous. Topological characterizations include: f is lsc iff for every α, the set {x : f(x) > α} is open; f is usc iff for every α, the set {x : f(x) < α} is open. In the extended real setting, f is lsc iff its epigraph {(x, t) : t ≥ f(x)} is closed, and f is usc iff its hypograph {(x, t) : t ≤ f(x)} is closed.

Examples: the indicator function of an open set is lower semicontinuous, while the indicator function of a closed set is upper semicontinuous. Continuous functions are both lsc and usc. Properties include that the sum of finitely many lsc functions is lsc, and the pointwise maximum of lsc functions is lsc (analogous statements hold for usc).

Applications appear in optimization, variational analysis, and convergence analysis, where semicontinuity conditions ensure existence of minima and well-behaved limits.