enow.com Web Search

Search results

  1. Results from the WOW.Com Content Network
  2. Discrete fixed-point theorem - Wikipedia

    en.wikipedia.org/wiki/Discrete_fixed-point_theorem

    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.

  3. Closed graph property - Wikipedia

    en.wikipedia.org/wiki/Closed_graph_property

    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 : XY be defined by f(0) = 1 and f(x) = 0 for all x ≠ 0. Then f : XY is continuous but its graph is not closed in X × Y. [4]

  4. Closed graph theorem (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/Closed_graph_theorem...

    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.)

  5. Closed graph theorem - Wikipedia

    en.wikipedia.org/wiki/Closed_graph_theorem

    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.

  6. Open and closed maps - Wikipedia

    en.wikipedia.org/wiki/Open_and_closed_maps

    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.

  7. Nowhere continuous function - Wikipedia

    en.wikipedia.org/wiki/Nowhere_continuous_function

    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 | () |.

  8. Fixed point (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Fixed_point_(mathematics)

    Let ≤ be a partial order over a set X and let f: XX be a function over X. Then a prefixed point (also spelled pre-fixed point, sometimes shortened to prefixpoint or pre-fixpoint) [citation needed] of f is any p such that f(p) ≤ p. Analogously, a postfixed point of f is any p such that p ≤ f(p). [3] The opposite usage occasionally ...

  9. Discrete mathematics - Wikipedia

    en.wikipedia.org/wiki/Discrete_mathematics

    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.