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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.
Weierstrass's function is also everywhere continuous but nowhere differentiable. ... Symmetric to the concept of a continuous map is an open map, ...
(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".
The compact-open topology on the set C(X, Y) of all continuous maps between two spaces X and Y is defined as follows: given a compact subset K of X and an open subset U of Y, let V(K, U) denote the set of all maps f in C(X, Y) such that f(K) is contained in U. Then the collection of all such V(K, U) is a subbase for the compact-open topology ...
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 ...
Continuous function: in which preimages of open sets are open. Nowhere continuous function: is not continuous at any point of its domain; for example, the Dirichlet function. Homeomorphism: is a bijective function that is also continuous, and whose inverse is continuous. Open function: maps open sets to open sets.
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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.