﻿ Semi-continuity - 计算数学达人 - 专，学者，数值代数，微分方程数值解

# Semi-continuity-only for study

For the notion of upper or lower semicontinuous multivalued function see: Hemicontinuity

In mathematical analysis, semi-continuity (or semicontinuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semi-continuous at a point x0 if, roughly speaking, the function values for arguments near x0 are either close to f(x0) or less than (respectively, greater than) f(x0).

[hide]

## Examples

Consider the function f, piecewise defined by f(x) = –1 for x < 0 and f(x) = 1 for x ≥ 0. This function is upper semi-continuous at x0 = 0, but not lower semi-continuous.

The indicator function of an open set is lower semi-continuous, whereas the indicator function of a closed set is upper semi-continuous. The floor function $f(x)=lfloor x rfloor$, which returns the greatest integer less than or equal to a given real number x, is everywhere upper semi-continuous. Similarly, the ceiling function $f(x)=lceil x rceil$ is lower semi-continuous.

A function may be upper or lower semi-continuous without being either left or right continuous. For example, the function $f(x) = begin{cases} 1 & mbox{if } x < 1, 2 & mbox{if } x = 1, 1/2 & mbox{if } x > 1, end{cases}$

is upper semi-continuous at x = 1 although not left or right continuous. The limit from the left is equal to 1 and the limit from the right is equal to 1/2, both of which are different from the function value of 2. Similarly the function $f(x) = begin{cases} sin(1/x) & mbox{if } x neq 0, 1 & mbox{if } x = 0, end{cases}$

is upper semi-continuous at x = 0 while the function limits from the left or right at zero do not even exist.

If $X=mathbb R^n$ is an Euclidean space (or more generally, a metric space) and $Gamma=C([0,1],X)$ is the space of curves in $X$ ( with the supremum distance $d_Gamma(alpha,beta)=sup_t d_X(alpha(t),beta(t))$, then the length functional $L:Gammato[0,+infty]$, which assigns to each curve $alpha$ its length $L(alpha)$, is lower semicontinuous.

Let $(X,mu)$ be a measure space and let $L^+(X,mu)$ denote the set of positive measurable functions endowed with the topology of $mu$-almost everywhere convergence.[dubious] Then the integral, seen as an operator from $L^+(X,mu)$ to $[-infty,+infty]$ is lower semi-continuous. This is just Fatou's lemma.

## Formal definition

Suppose X is a topological space, x0 is a point in X and f : XR ∪ {–∞,+∞} is an extended real-valued function. We say that f is upper semi-continuous at x0 if for every ε > 0 there exists a neighborhoodU of x0 such that f(x) ≤ f(x0) + ε for all x in U. For the particular case of a metric space, this can be expressed as $limsup_{xto x_{0}} f(x)le f(x_0)$

where lim sup is the limit superior (of the function f at point x0). (For non-metric spaces, an equivalent definition using nets may be stated.)

The function f is called upper semi-continuous if it is upper semi-continuous at every point of its domain. A function is upper semi-continuous if and only if {xX : f(x) < α} is an open set for every α ∈ R.

We say that f is lower semi-continuous at x0 if for every ε > 0 there exists a neighborhoodU of x0 such that f(x) ≥ f(x0) – ε for all x in U. Equivalently, this can be expressed as $liminf_{xto x_0} f(x)ge f(x_0)$

where lim inf is the limit inferior (of the function f at point x0).

The function f is called lower semi-continuous if it is lower semi-continuous at every point of its domain. A function is lower semi-continuous if and only if {xX : f(x) > α} is an open set for every α ∈ R; alternatively, a function is lower semi-continuous if and only if all of its lower levelsets {xX : f(x) ≤ α} are closed. Lower level sets are also called sublevel sets or trenches.

## Properties

A function is continuous at x0 if and only if it is upper and lower semi-continuous there. Therefore, semi-continuity can be used to prove continuity.

If f and g are two real-valued functions which are both upper semi-continuous at x0, then so is f + g. If both functions are non-negative, then the product function fg will also be upper semi-continuous at x0. Multiplying a positive upper semi-continuous function with a negative number turns it into a lower semi-continuous function.

If C is a compact space (for instance a closed, boundedinterval [a, b]) and f : C → [–∞,∞) is upper semi-continuous, then f has a maximum on C. The analogous statement for (–∞,∞]-valued lower semi-continuous functions and minima is also true. (See the article on the extreme value theorem for a proof.)

Suppose fi : X → [–∞,∞] is a lower semi-continuous function for every index i in a nonempty set I, and define f as pointwise supremum, i.e., $f(x)=sup_{iin I}f_i(x),qquad xin X.$

Then f is lower semi-continuous. Even if all the fi are continuous, f need not be continuous: indeed every lower semi-continuous function on a uniform space (e.g. a metric space) arises as the supremum of a sequence of continuous functions.

The indicator function of any open set is lower semicontinuous. The indicator function of a closed set is upper semicontinuous. However, in convex analysis, the term "indicator function" often refers to the characteristic function, and the characteristic function of any closed set is lower semicontinuous, and the characteristic function of any open set is upper semicontinuous.

A function f : RnR is lower semicontinuous if and only if its epigraph (the set of points lying on or above its graph) is closed.

A function f : XR, for some topological space X, is lower semicontinuous if and only if it is continuous with respect to the Scott topology on R.

## 