Questions tagged [continuity]
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17
questions
33
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What is the smallest set of real continuous functions generating all rational numbers by iteration?
I recently came across this problem from USAMO 2005:
"A calculator is broken so that the only keys that still
work are the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$ and $\arctan$ buttons. The ...
- 4,792
32
votes
2
answers
2k
views
Quantifier complexity of the definition of continuity of functions
This was previously asked at MSE, but I was told to ask it on MO. Consider the structure $(\mathbb{R};+,-,*,0,1,<)$. We adjoin to it a unary function $f$ defined everywhere on the set of real ...
- 2,063
13
votes
3
answers
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What is the definition of continuity of set-valued functions?
According to the wiki of Kakutani's fixed-point theorem, A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W \...
- 133
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votes
3
answers
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Are measurable homomorphisms $ (\Bbb{C},+) \to (\Bbb{C},+) $ or $ (\Bbb{C},+) \to (\Bbb{C},*) $ continuous, and do they admit an explicit description?
I am interested in generalizations of the following fact (known as automatic continuity, as pointed out below). I am especially looking for references to papers dating back to 1920’s. I feel that ...
- 1,299
9
votes
2
answers
626
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Is $\mathbb{Q}$ the orbit of a rational function under iteration?
In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb Q$ by iteration starting from $0$. Surprisingly one continuous function suffices.
In the ...
- 4,792
8
votes
2
answers
741
views
Are point sets of the same order type connected by continuous (order type)-preserving motion?
Given two general position point sets in $\mathbb{R}^2$ of the same size and order type, is it possible to continuously move the points of one set until they coincide with those of the other set in ...
- 83
7
votes
0
answers
225
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Uniform approximation of separately continuous functions on zero-dimensional spaces
For topological spaces $X,Y,Z$ а function $f:X\times Y\to Z$ is called separately continuous if for any $(x,y)\in X\times Y$ the restrictions of $f$ to the sets $\{x\}\times Y$ and $X\times \{y\}$ are ...
- 40.9k
6
votes
1
answer
341
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Quantifier complexity of definition of compactness
This question is inspired by the post on quantifier complexity of
continuity. We work with metric spaces M
considered as two-sorted first-order structures (M,$\mathbb R$,d,+,⋅,<)
where $d:M^2→\...
- 15.2k
5
votes
2
answers
375
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Does convergence in law to absolutely continuous limit imply convergence in convex distance?
Let $(X_n)$ be a sequence of $\mathbb{R}^d$-valued random variables converging in distribution to some limiting random variable $X$ whose CDF is absolutely continuous with respect to the Lebesgue ...
- 617
5
votes
1
answer
216
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"Uniformly continuous" environment sum of a bijection $\varphi:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}$
Given any function $f: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$ we define the environment sum of $(x,y)\in\mathbb{Z}\times \mathbb{Z}$ with respect to $f$ by
$$\text{es}_f(x,y) = \sum\{f(x', y'): |(...
- 45.4k
4
votes
4
answers
606
views
Continuity in Banach space for non-linear maps
I want to find an example of a Banach space $X$ and a continuous map $f:X\rightarrow X$ such that $f$ is not bounded on the unit ball. I do not doubt that such an example exists, but I cannot make it ...
- 15.2k
3
votes
1
answer
303
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Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?
Say we have a power series of two variables, with an associated function $f$ defined as
$$
\begin{split}
f(x, y) =\, & \sum_{n,m} a_{n,m}x^ny^m,\\
& a_{n,m} \geq 0 \quad \forall n, m \in\...
- 63
3
votes
1
answer
696
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Continuity of a parameterized convex optimization problem
I have a parameterised optimization problem:
\begin{align}
\boldsymbol{S}(p)= &\arg \min_{ \boldsymbol{x}} g( \boldsymbol{x})\\
\text{s.t. } & \boldsymbol{A}(p) \textbf{x} = \...
- 88
3
votes
1
answer
547
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upper semicontinuity in C(X)-algebras
I've stuck on a step of proposition 1.2 of Rieffel's article (continuous field of C*-algebras coming from group cocycles and actions, 1989). I think it basically proves that a C(X)-algebras coming ...
- 342
1
vote
1
answer
218
views
Continuity of the solution of a Pde system
Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$ both continuous and bounded.
I have the following system of PDE's
\begin{align}
\begin{cases}
\frac{\partial}{\partial t} u_0(t,r)=- J* ...
- 139
1
vote
0
answers
740
views
A weakly sequentially continuous operator which is not weakly continuous
I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.
So, let
$T$ an operator between a Banach space $X$ and itself.
$T$ is weakly ...
- 291
1
vote
1
answer
225
views
Continuity of $r\mapsto\int_{\Sigma\cap B_r(x)}f^2d\mu$
Let $\Sigma$ be an embedded smooth surface in $\mathbb{R}^3$, and let $f:\Sigma\to\mathbb{R}$ be a smooth function. Suppose $f$ is square-integrable on $\Sigma$, with
\begin{align}
0<\int_{\Sigma}f^...