All Questions
Tagged with metric-spaces continuity
8 questions
5
votes
1
answer
373
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Points of differentiability of squared distance from a point in metric spaces
I posted this same question on MSE with no answer.
Let $I:=(0, + \infty)$ and let $(X,d)$ be a complete and separable metric space. In this setting we say that $u : I \to X$ is absolutely continuous ...
- 401
4
votes
1
answer
874
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Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?
Recently I came to know about Atsuji space from the paper1. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have ...
- 195
2
votes
1
answer
79
views
Hausdorff-Lipschitz continuity of cone correspondence
Let $\mathbb{R}_+$ denote the strictly positive real numbers, let $\mathcal{X} \subset \mathbb{R}^n$ and $\mathcal{P} \subset \mathbb{R}^m$ be compact and convex subsets, let
\begin{equation}
f: \...
- 55
1
vote
2
answers
484
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Is there good evidence that topological spaces are the correct way to study the general theory of continuity? [closed]
My reason for asking is that the theory of metric spaces is so clean and so many significant theorems can be proved for an arbitrary metric space (which makes it plausible to me that metric spaces are ...
- 4,351
1
vote
1
answer
132
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Variants of Dirichlet-type function as a pointwise limit of continuous functions
Problem
Suppose $f$ is a function from a complete metric space $X$ to a metric space $Y$, and suppose $Y$ has points $y_{0}$, $y_{1}$ such that the subsets $f^{-1}(y_{0})$ and $f^{-1}(y_{1})$ are both ...
- 129
1
vote
0
answers
165
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Uniformly open map on a dense subset
Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion.
I think the ...
- 11
0
votes
1
answer
268
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Topologies and Borel $\sigma$-fields on disjoint unions
Consider a set of functions $\mathcal{F}$ on $E$ where $E \subset\mathbb{R}^k$ - e.g. the class of $L_1$ functions on $[0,1]$ - and endow it with a suitable metric $d$ that makes it Polish.
Consider ...
- 195
0
votes
1
answer
275
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An extension for lower semi continuous lower bounded real valued functions class
Let $(X,d)$ be a complete metric space. I need some explanations about the class of all functions like $f$ which have $f:X \to \mathbb{R}\cup\{ +\infty\}$, be a lower bounded and, for all $y \in X$ we ...
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