### Semi-continuity

作者：niuman 发布时间：2013-02-04 12:17:33 浏览次数：692# Semi-continuity-only for study

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*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 *x*_{0} if, roughly speaking, the function values for arguments near *x*_{0} are either close to *f*(*x*_{0}) or less than (respectively, greater than) *f*(*x*_{0}).

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## [edit]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 *x*_{0} = 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, which returns the greatest integer less than or equal to a given real number *x*, is everywhere upper semi-continuous. Similarly, the ceiling function is lower semi-continuous.

A function may be upper or lower semi-continuous without being either left or right continuous. For example, the function

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

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

If is an Euclidean space (or more generally, a metric space) and is the space of curves in ( with the supremum distance, then the length functional , which assigns to each curve its length, is lower semicontinuous.

Let be a measure space and let denote the set of positive measurable functions endowed with the topology of -almost everywhere convergence.^{[dubious– discuss]} Then the integral, seen as an operator from to is lower semi-continuous. This is just Fatou's lemma.

## [edit]Formal definition

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

where lim sup is the limit superior (of the function *f* at point *x*_{0}). (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 {*x* ∈ *X* : *f*(*x*) < α} is an open set for every α ∈ **R**.

We say that *f* is **lower semi-continuous** at *x*_{0} if for every ε > 0 there exists a neighborhood*U* of *x*_{0} such that *f*(*x*) ≥ *f*(*x*_{0}) – ε for all *x* in *U*. Equivalently, this can be expressed as

where lim inf is the limit inferior (of the function *f* at point *x*_{0}).

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 {*x* ∈ *X* : *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 {*x* ∈ *X* : *f*(*x*) ≤ α} are closed. Lower level sets are also called *sublevel sets* or *trenches*.^{[1]}

## [edit]Properties

A function is continuous at *x*_{0} 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 *x*_{0}, then so is *f* + *g*. If both functions are non-negative, then the product function *fg* will also be upper semi-continuous at *x*_{0}. 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 *f*_{i} : *X* → [–∞,∞] is a lower semi-continuous function for every index *i* in a nonempty set *I*, and define *f* as pointwise supremum, i.e.,

Then *f* is lower semi-continuous. Even if all the *f*_{i} 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* : **R**^{n}→**R** is lower semicontinuous if and only if its epigraph (the set of points lying on or above its graph) is closed.

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

## [edit]References

Bourbaki, Nicolas (1998).

*Elements of Mathematics: General Topology, 1–4*. Springer. ISBN 0-201-00636-7.Bourbaki, Nicolas (1998).

*Elements of Mathematics: General Topology, 5–10*. Springer. ISBN 3-540-64563-2.Gelbaum, Bernard R.; Olmsted, John M.H. (2003).

*Counterexamples in analysis*. Dover Publications. ISBN 0-486-42875-3.Hyers, Donald H.; Isac, George; Rassias, Themistocles M. (1997).

*Topics in nonlinear analysis & applications*. World Scientific. ISBN 981-02-2534-2.

## [edit]See also