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Semi-continuity

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In mathematical analysis, semi-continuity (or semicontinuity) is a property of extended real-valued functions that is weaker than continuity. [1]

35 relations: Bounded set, Characteristic function (convex analysis), Closed set, Compact space, Continuous function, Convergence in measure, Curve, Dense set, Domain of a function, Epigraph (mathematics), Extended real number line, Extreme value theorem, Fatou's lemma, Floor and ceiling functions, Function (mathematics), Function composition, Graph of a function, Hemicontinuity, Indicator function, Infimum and supremum, Interval (mathematics), Level set, Limit superior and limit inferior, Mathematical analysis, Mathematical optimization, Metric space, Multivalued function, Neighbourhood (mathematics), Net (mathematics), Open set, Piecewise, Scott continuity, Topological space, Uniform norm, Uniform space.

Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.

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Characteristic function (convex analysis)

In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set.

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Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

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Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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Convergence in measure

Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability.

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Curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.

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Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).

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Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

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Epigraph (mathematics)

In mathematics, the epigraph or supergraph of a function f: Rn→R is the set of points lying on or above its graph: The strict epigraph is the epigraph with the graph itself removed: The same definitions are valid for a function that takes values in R ∪ ∞.

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Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system by adding two elements: and (read as positive infinity and negative infinity respectively).

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Extreme value theorem

In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval, then f must attain a maximum and a minimum, each at least once.

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Fatou's lemma

In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions.

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Floor and ceiling functions

In mathematics and computer science, the floor function is the function that takes as input a real number x and gives as output the greatest integer less than or equal to x, denoted \operatorname(x).

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

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Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

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Hemicontinuity

In mathematics, the notion of the continuity of functions is not immediately extensible to multivalued mappings or correspondences between two sets A and B. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension.

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Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.

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Infimum and supremum

In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.

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Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

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Level set

In mathematics, a level set of a real-valued function ''f'' of ''n'' real variables is a set of the form that is, a set where the function takes on a given constant value c. When the number of variables is two, a level set is generically a curve, called a level curve, contour line, or isoline.

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Limit superior and limit inferior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence.

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Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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Mathematical optimization

In mathematics, computer science and operations research, mathematical optimization or mathematical programming, alternatively spelled optimisation, is the selection of a best element (with regard to some criterion) from some set of available alternatives.

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Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

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Multivalued function

In mathematics, a multivalued function from a domain to a codomain is a heterogeneous relation.

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Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

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Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence.

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Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

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Piecewise

In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain, a sub-domain.

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Scott continuity

In mathematics, given two partially ordered sets P and Q, a function f \colon P \rightarrow Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema, i.e. if for every directed subset D of P with supremum in P its image has a supremum in Q, and that supremum is the image of the supremum of D: that is, \sqcup f.

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Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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Uniform norm

In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number This norm is also called the supremum norm, the Chebyshev norm, or the infinity norm. The name "uniform norm" derives from the fact that a sequence of functions \ converges to f under the metric derived from the uniform norm if and only if f_n converges to f uniformly.

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Uniform space

In the mathematical field of topology, a uniform space is a set with a uniform structure.

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Lower hemicontinuous, Lower semi-continuous, Lower semicontinuous, Semi-continuous, Semi-continuous function, Semi-continuous mapping, Semicontinuity, Semicontinuous, Semicontinuous function, Upper semi-continuous, Upper semicontinuous, Upper-semicontinuous.

References

[1] https://en.wikipedia.org/wiki/Semi-continuity

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