The continuity tag has no wiki summary.
1
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1answer
112 views
Follow up: Is continuity preserved when taking comma object?
Here, I asked wether taking lax pullback preserves continuity, but got no precise answer.
However, I have found this recent article by Riehl and Verity which proves something very similar, but I ...
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votes
0answers
16 views
Show that uniform continuity implies stochastic equicontinuity
Let $\Theta$ be a metric space and assume it is compact. Let $W_t: \Omega \rightarrow \mathbb{R}^k$ be a random variable for $t\leq T$. Let $m(.,\theta): \mathbb{R}^k\rightarrow\mathbb{R}^s$. Let ...
1
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0answers
92 views
Lipschitz continuity of solution set mapping of a parametric convex optimization problem
I have a parametric convex optimization problem:
\begin{array}{cl}
\underset{x}{\text{minimize}} & f\left(x,z\right)\\
\text{subject to} & g\left(x\right)\leq0
\end{array}
where $x$ is the ...
2
votes
0answers
78 views
How are measurable functions morphisms?
I am trying to encode the theory of measurable sets in higher order logic. I already did so with the theory of topology. I think it is relevant because it enables one to see continuous or measurable ...
7
votes
1answer
266 views
Extending a $ * $-Representation of $ ({C_{c}}(G,\mathscr{A}),\star,^{*}) $ to a $ * $-Representation of $ \mathscr{A} \rtimes_{\alpha} G $
Let $ (\mathscr{A},G,\alpha) $ be a $ C^{*} $-dynamical system, and consider the twisted convolution $ * $-algebra $ ({L^{1}}(G,\mathscr{A}),\star,^{*}) $ defined by
\begin{align*}
\forall \phi,\psi ...
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0answers
23 views
B-spline re-parametrization
I have a question regarding the re-parameterisation of a B-spline.
Some info:
The B-spline is of order 4 (degree 5), hence $C^3$ continuity
There is no knot multiplicity
The end conditions are not ...
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0answers
56 views
Continuity of the real Monge Ampère operator on convex functions
Let $E\subset\mathbb{R}^n$ be a convex set, $u:E\to\mathbb{R}$ a convex function and $B\subset E$ a Borel set.
We define the (multivalued) gradient
\begin{array}{rccl}
\nabla [u]:& \mathbb{R}^n ...
7
votes
2answers
561 views
the convolution of integrable functions is continuous?
The question is simple but I still can't prove it or contradict it. Here it goes:
Suppose $f$ and $g$ are defined on the circle
(or, equivalently, $2\pi$ periodic functions) and Lebesgue ...
2
votes
3answers
178 views
Speed of convergence for Weyl's Equidistribution theorem
If $f$ is a continuous periodic fonction on $[0,1]$ and $a\not\in\mathbb{Q}$, the Weyl's equidistribution theorem states that
$$\frac{1}{n}\sum_{k=0}^{n-1}f(ak)\rightarrow \int_0^1 f(x)dx.$$
Can we ...
5
votes
3answers
635 views
On the continuity of $\sum_{n=1}^{\infty} sin(nx) / n^\alpha$
I know that the series $\sum_{n=1}^\infty \sin(nx) / n^\alpha$, with $0 < \alpha < 1$, converges for $x \in [0,2\pi]$.
I'm trying to understand if the function $f(x) = \sum_{n=1}^\infty ...
2
votes
1answer
123 views
Find a continuous function with a prescribed continuity set
It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$.
In the book "Understanding Analysis" by Abbott is stated in page 128 ...
7
votes
3answers
719 views
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 ...
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0answers
175 views
Continuity of a function
Let $f\in L^2(\mathbb{R}^3)$ with compact suppport and $z\in\mathbb{C}$. Is the following function continuous for $z\in Q = \{ z : \Re z\in [a,b], \Im \sqrt{z} \in (0,1] \}$:
$$ ...
2
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0answers
58 views
Presentation of tree decompositions (and related concepts) in terms of continuous maps?
A tree decomposition of a graph $G$ is commonly defined in terms of a tree $T$ with the following structure:
Each vertex $t \in V(T)$ is associated to a set $X_t \subseteq V(G)$;
The union ...
0
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1answer
357 views
Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable?
Changing my question in light of Dan's answer. Thanks, Dan.
Consider a sequence of real random variables $X_i$ bounded in $L_1$, that is $\mathbb E\left|X_i\right|\leq M$ for all $i$. Suppose that ...
3
votes
2answers
103 views
continuity of length function $l: T(X) \times MF \to \mathbb R$
Let $T(X)$ be the Teichmuller space of a closed Riemann surface $X$ of genus $g \geq 2,$ and $MF$ the space of equivalence classes of measured foliations. Then we have a length function
$l: T(X) ...
3
votes
1answer
219 views
Automatic continuity of the inverse map
All topological spaces considered here are Hausdorff.
It is a well-known consequence of the minimality of a compact topology that an injective continuous map
$f\colon X\to Y$
where $X$ is compact, ...
0
votes
1answer
93 views
Continuity of critical points with respect to a parameterisation.
Hello.
I have a research note coming out soon, and I'm stuck showing that a weird kind of function is continuous. I need to it show a new method of bounding exponential growth factors in ...
1
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0answers
88 views
Is the Poincare action on the Klein-Gordon quantum field strongly continuous?
I am interested in checking continuity property of the Poincare group action on the Klein-Gordon quantum field theory defined over the Minkowski spacetime. Maybe the simplest example of QFT out there.
...
1
vote
2answers
496 views
Criteria for Lipschitz continuity
Is the following statement true. Assume that $f:[0,1]\to [0,1]$ is a continuous function such that $$\sup_t\lim\sup_{s\to t}\frac{|f(s)-f(t)|}{|t-s|}<\infty,$$ then $f$ is Lipchitz continuous.
2
votes
1answer
205 views
Function spaces over pseudocompact spaces
Let $K$ be a pseudocompact Tychonoff space that is, a Hausdorff $T_{3.5}$ space for which every continuous function $\varphi \colon K\to \mathbb{C}$ is bounded. Let $\beta$ be the Stone-Cech extension ...
2
votes
1answer
242 views
upper semicontinuity in C(X)-algebras
Dear fellows,
I've stuck on a step of proposition 1.2 of Rieffel's article (continuous field of C*-algebras coming from group cocycles and actions, 1989). I think it basically proves that a ...
4
votes
1answer
232 views
Are point sets of the same order type connected by continuous (order type)-preserving motion?
Given two general position point sets in $\mathbb{R}^2$ of the same size and order type, is it possible to continuously move the points of one set until they coincide with those of the other set in ...
0
votes
1answer
113 views
Continuity of Lexicographic Minimum Solution of a parametrized LP problem
Given a parametrized LP problem
find x, that minimizes F*x
such that Ax <=Bt+D
where t is a parameter.
And suppose C(t) is a set of all optimal solutions of LP with parameter t.
Let x_L(t) be ...
4
votes
2answers
241 views
Lipschitz continuity of singular values
How smooth are the singular values of a matrix F in terms of entries of F? I am hoping for Lipschitz continuity, but was not able to find it.
5
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0answers
197 views
Topological conditions forcing continuity
Let $X$, $Y$, and $Z$ be topological spaces. Let $f:X \rightarrow Y$. Further assume that for every continuous function $g:Y \rightarrow Z$, $g \circ f$ is continuous.
Question: Under what ...
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4answers
2k views
Does Cauchy continuity imply uniform continuity? [No.] [closed]
It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff
$$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X ...
