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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. 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}.

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 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

- $f(x)\; =\; begin\{cases\}$

- $f(x)\; =\; begin\{cases\}$

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. Equivalently, 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 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 ∈ 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

- $liminf\_\{xto\; x\_0\}\; f(x)ge\; f(x\_0)$

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 ∈ X : f(x) > α} is an open set for every α ∈ R.

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 → [–∞,∞) 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.,

- $f(x)=sup\_\{iin\; I\}f\_i(x),qquad\; xin\; X.$

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.

- Bourbaki, Nicolas
*Elements of Mathematics: General Topology, 1–4*. Springer. - Bourbaki, Nicolas
*Elements of Mathematics: General Topology, 5–10*. Springer. - Gelbaum, Bernard R.; Olmsted, John M.H.
*Counterexamples in analysis*. Dover Publications. - Hyers, Donald H.; Isac, George; Rassias, Themistocles M.
*Topics in nonlinear analysis & applications*. World Scientific.

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Last updated on Wednesday August 27, 2008 at 17:36:10 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday August 27, 2008 at 17:36:10 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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