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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 | () |.
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]
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.)
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.
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic.
It turns out that the Weierstrass function is far from being an isolated example: although it is "pathological", it is also "typical" of continuous functions: In a topological sense: the set of nowhere-differentiable real-valued functions on [0, 1] is comeager in the vector space C ([0, 1]; R ) of all continuous real-valued functions on [0, 1 ...
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.