Semi-continuity

:"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. A extended real-valued function "f" is upper 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 "f"("x"₀). If "less than" is replaced by "greater than", the function is called lower semi-continuous at "x"₀.

Examples

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

The floor function $f(x)=lfloor\; x\; floor$, which returns the greatest integer less than or equal to a given real number "x", is everywhere upper semi-continuous. Similarly, the ceiling function $f(x)=lceil\; x\; ceil$ 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.

Formal definition

Suppose "X" is a topological space, "x"₀ is a point in "X" and "f" : "X" &rarr; R &cup; {–∞,+∞} is an extended real-valued function. We say that "f" is upper semi-continuous at "x"₀ if for every &epsilon; > 0 there exists a neighborhood "U" of "x"₀ such that "f"("x") ≤ "f"("x"₀) + &epsilon; for all "x" in "U". Equivalently, this can be expressed as

:$limsup\_\{x\; o\; x\_0\}\; f(x)le\; f(x\_0)$

where lim sup is the limit superior (of the function "f" at point "x"₀).

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" &isin; "X" : "f"("x") &lt; α} is an open set for every α &isin; R.

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

:$liminf\_\{x\; o\; x\_0\}\; f(x)ge\; f(x\_0)$

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" &isin; "X" : "f"("x") &gt; α} is an open set for every α &isin; R.

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, bounded interval ["a", "b"] ) and "f" : "C" &rarr; [–∞,∞) 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" &rarr; [–∞,∞] 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 "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.

References

*cite book
last = Bourbaki
first = Nicolas
title = Elements of Mathematics: General Topology, 1–4
publisher = Springer
date = 1998
pages =
isbn = 0201006367
*cite book
last = Bourbaki
first = Nicolas
title = Elements of Mathematics: General Topology, 5–10
publisher = Springer
date = 1998
pages =
isbn = 3540645632
*cite book
last = Gelbaum
first = Bernard R.
coauthors = Olmsted, John M.H.
title = Counterexamples in analysis
publisher = Dover Publications
date = 2003
pages =
isbn = 0486428753
*cite book
last = Hyers
first = Donald H.
coauthors = Isac, George; Rassias, Themistocles M.
title = Topics in nonlinear analysis & applications
publisher = World Scientific
date = 1997
pages =
isbn = 9810225342

ee also

* Directional continuity
* Semicontinuous multivalued function