Search results
Results From The WOW.Com Content Network
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 ...
The graph of a cubic function has no jumps or holes. The function is continuous. Checking the continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function.
The closed graph theorem is an important result in functional analysis that guarantees that a closed linear operator is continuous under certain conditions. The original result has been generalized many times. A well known version of the closed graph theorems is the following.
Closed graph theorems are of particular interest in functional analysis where there are many theorems giving conditions under which a linear map with a closed graph is necessarily continuous. If f : X → Y is a function between topological spaces whose graph is closed in X × Y and if Y is a compact space then f : X → Y is continuous.
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.)
The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.
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
For uniformly continuous functions, for each positive real number > there is a positive real number > such that when we draw a rectangle around each point of the graph with a width slightly less than and a height slightly less than , the graph lies completely inside the height of the rectangle.