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See the page on direction-preserving function for definitions. Continuous fixed-point theorems often require a convex set. The analogue of this property for discrete sets is an integrally-convex set. A fixed point of a discrete function f is defined exactly as for continuous functions: it is a point x for which f(x)=x.
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.)
Let X denote the real numbers ℝ with the usual Euclidean topology and let Y denote ℝ with the indiscrete topology (where note that Y is not Hausdorff and that every function valued in Y is continuous). Let f : X → Y be defined by f(0) = 1 and f(x) = 0 for all x ≠ 0. Then f : X → Y is continuous but its graph is not closed in X × Y. [4]
A real function that is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition is given below.
In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain.If is a function from real numbers to real numbers, then is nowhere continuous if for each point there is some > such that for every >, we can find a point such that | | < and | () |.
This function from the unit circle to the half-open interval [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of a compact space under an open or closed map need not be compact. Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed.
For functions that are not uniformly continuous, this isn't possible; for these functions, the graph might lie inside the height of the rectangle at some point on the graph but there is a point on the graph where the graph lies above or below the rectangle.