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A linear map between two topological vector spaces, such as normed spaces for example, is continuous (everywhere) if and only if there exists a point at which it is continuous, in which case it is even uniformly continuous. Consequently, every linear map is either continuous everywhere or else continuous nowhere.
Choosing an orientation means choosing a generator of the top homology group. A continuous map f : X →Y induces a homomorphism f ∗ from H m (X) to H m (Y). Let [X], resp. [Y] be the chosen generator of H m (X), resp. H m (Y) (or the fundamental class of X, Y). Then the degree of f is defined to be f * ([X]). In other words,
In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems.
For example, H. G. Garnir, in searching for so-called "dream spaces" (topological vector spaces on which every linear map into a normed space is continuous), was led to adopt ZF + DC + BP (dependent choice is a weakened form and the Baire property is a negation of strong AC) as his axioms to prove the Garnir–Wright closed graph theorem which ...
Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. If an open map f has an inverse function , that inverse is continuous, and if a continuous map g has an inverse, that inverse is open.
More generally, a Blumberg space is a topological space for which any function : admits a continuous restriction on a dense subset of . The Blumberg theorem therefore asserts that (equipped with its usual topology) is a Blumberg space.
Pavel Urysohn. In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or Urysohn-Brouwer lemma [1]) states that any real-valued, continuous function on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
(a different Weierstrass Function which is also continuous and nowhere differentiable) Nowhere differentiable continuous function proof of existence using Banach's contraction principle. Nowhere monotonic continuous function proof of existence using the Baire category theorem. Johan Thim. "Continuous Nowhere Differentiable Functions".