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set-valued function with a closed graph. If F : X → 2 Y is a set-valued function between topological spaces X and Y then the following are equivalent: F has a closed graph (in X × Y); (definition) the graph of F is a closed subset of X × Y; and if Y is compact and Hausdorff then we may add to this list:
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. A blog post [1] by T. Tao lists several closed graph theorems throughout mathematics.
The usual proof of the closed graph theorem employs the open mapping theorem.It simply uses a general recipe of obtaining the closed graph theorem from the open mapping theorem; see closed graph theorem § Relation to the open mapping theorem (this deduction is formal and does not use linearity; the linearity is needed to appeal to the open mapping theorem which relies on the linearity.)
Homeomorphism (graph theory) – Concept in graph theory (closely related to graph subdivision) Homotopy#Isotopy – Continuous deformation between two continuous functions; Mapping class group – Group of isotopy classes of a topological automorphism group; Poincaré conjecture – Theorem in geometric topology; Universal homeomorphism
However, not all functions are continuous. If a function is not continuous at a limit point (also called "accumulation point" or "cluster point") of its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.
Function : is graph continuous if for all there exists a function : such that ((),) is continuous at .. Dasgupta and Maskin named this property "graph continuity" because, if one plots a graph of a player's payoff as a function of his own strategy (keeping the other players' strategies fixed), then a graph-continuous payoff function will result in this graph changing continuously as one varies ...
An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions : to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in which the only open subsets are the empty set and X ) and the space T set is at least T 0 , then the only continuous ...
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.