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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.
Every continuous function from a closed ball of a Euclidean space into itself has a fixed point. [7] A slightly more general version is as follows: [8] Convex compact set Every continuous function from a nonempty convex compact subset K of a Euclidean space to K itself has a fixed point. [9] An even more general form is better known under a ...
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.
(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".
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.
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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 ...
Downey shrugs at his costar Woody Harrelson, who smirks.Harrelson is bald, having shaved his head two weeks ago on camera after Stone declared that his character, a serial killer, should shave his ...