Theory of continuous functions

function such that the preimage of an open set is open

In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.

theorem

thumb|The two dashed Path (topology)|paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy. In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

property limiting the "growth" of distances of outputs of a function uniformly across its domain

function that is continuous everywhere but differentiable nowhere

every continuous function on a compact set has a fixed point

a particular example of a space-filling curve, discovered by Giuseppe Peano

problem of oscillation at the edges of an interval when using polynomial interpolation

point at which a function is not continuous

theorem

theorem

lemma that a topological space is normal iff any 2 disjoint closed subsets can be separated by a continuous function

theorem that every continuous function on a compact Hausdorff space can be approximated by certain families of continuous functions

form of continuity for functions

thumb|right|An upper semicontinuous function that is not lower semicontinuous at x_0. The solid blue dot indicates f\left(x_0\right). thumb|right|A lower semicontinuous function that is not upper semicontinuous at x_0. The solid blue dot indicates f\left(x_0\right).

In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.

theorem

curve whose image is dense within an open region of the plane

linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X

theorem in real analysis

topological space consisting of equivalence classes of points in another topological space

theorem that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary

theorem

fractal-like continuous function that maps quadratic irrationals to rationals, and rationals to dyadic rationals

generalization of winding number

function between metric spaces that does not increase any distance

fractal which is considered to resemble a blancmange

theorem in topology about homeomorphic subsets of Euclidean space

mathematical function revertible near each point

continuous map such that preimages of compact sets are compact

Function between topological vector spaces

collection of continuous functions

integral transform

in algebraic topology, a theorem that a continuous map between (geometric realizations of) simplicial complexes is homotopy-equivalent to a simplicial map between subdivisions of the simplicial complexes