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In mathematics, a property of functions and their graphs. A continuous function is one whose graph has no breaks, gaps, or jumps. It is defined using the concept of a limit. Specifically, a function is said to be continuous at a value *math.x* if the limit of the function exists there and is equal to the function's value at that point. When this condition holds true for all real number values of *math.x* in an interval, the result is a graph that can be drawn over that interval without lifting the pencil. Such functions are crucial to the theory of calculus, not just because they model most physical systems but because the theorems that lead to the derivative and the integral assume the continuity of the functions involved.

Learn more about continuity with a free trial on Britannica.com.

Encyclopedia Britannica, 2008. Encyclopedia Britannica Online.

- 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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