Semicontinuous Function

Semicontinuous Function

 

a concept in mathematical analysis. A lower semicontinuous function at a point x0 is a function such that

Semicontinuous Function

Correspondingly for an upper function,

Semicontinuous Function

In other words, a function is lower semicontinuous at x0 if for every ε > 0 a number δ > 0 can be found such that ǀxx0ǀ < δ implies f(x0) — f(x) < ε (not in absolute value!).

A function that is both upper and lower semicontinuous is continuous in the usual sense. A number of properties of semi-continuous functions are analogous to those of continuous functions. For example, if f(x) and g(x) are lower semicontinuous, their sum and product are also lower semicontinuous; likewise, a lower semicontinuous function on an interval attains its minimum. If the functions un (x), n = 1,2, …, are lower semicontinuous and ≥0, the sum

Semicontinuous Function

is lower semicontinuous. Semicontinuous functions are Baire functions of class 1.

References in periodicals archive ?
Let [eta] : K x K [right arrow] R [union] {+[infinity]} be a proper convex lower semicontinuous function and [eta] : K x K [right arrow] E be a mapping.
N] [right arrow] (-[infinity], +[infinity]] be a proper and lower semicontinuous function.
If in Theorem 1 A is the subdifferential of a proper, convex and lower semicontinuous function [phi] : H [right arrow] (-[infinity], +[infinity]] and F is nonempty (i.
2]) for fixed u, it is an upper semicontinuous function of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
where f : E [right arrow] E is a self mapping and [phi] : [0, + [infinity]) [right arrow] [0, + [infinity]) is a lower semicontinuous function from right such that [phi] is positive on (0, + [infinity]) and [phi](0) = 0.
n], [rho] is a positive number, and [PHI] : [OMEGA] [right arrow] R [union] {+[infinity]} is a proper, convex, and lower semicontinuous function.
In [7], when A = [partial derivative][phi] where [phi] is a proper, convex and lower semicontinuous function, we proved an ergodic theorem and a weak convergence theorem for solutions to (1), by assuming (2), (3), (4) and that [t.
Let [PSI] : [omega] [right arrow] RU{+[infinity]} be a lower semicontinuous function.
From convex analysis it is well know that a proper, convex and lower semicontinuous function g: X [right arrow] [bar.
Let (M,[rho]) be a complete metric space and let [phi]: M [right arrow] R be a lower semicontinuous function which is bounded from below.
Let us assume that H is a lower semicontinuous function on [partial derivative][OMEGA].
lower) semicontinuous functions on L are upper (resp.