The continuity tag has no wiki summary.
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Oscillation Index of a function which set of discontinuities is countable
In this paper, the authors defined an oscillation index to be $\beta(f)=\sup_{\epsilon >0}{\beta(f,\epsilon)}$, which has something to do with the set of discontinuities of $f$. Also, the author ...
3
votes
1answer
86 views
If $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, does $f(u_n) \to f(u)$ in $H^{\frac 12}(\partial\Omega)$ for $f$ Lipschitz?
Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with $|f'(x)| \leq C$ for all $x$ and $f(0)=0$.
Suppose $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, where $\Omega$ is a bounded domain of ...
0
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0answers
35 views
Uniform continuity in the first variable on compacts w.r.t the other variables for a continuous function
It is a notational doubt.
I have seen the following notation in a book for the continuous function $f$:
$f(.,.):\Bbb R^d \times \Bbb R^d \to \Bbb R$ is uniformly continuous in the first argument on ...
0
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0answers
27 views
continuous extensions of concave functions
Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following:
Does a continuous and concave function
\begin{eqnarray*}
f: N_{\mathbb{Q}} \to \mathbb{R}
...
1
vote
1answer
123 views
Absolutely continuous functions
it is well known that if a function $f:[0,T]\to\mathbb{R}$ satisfies the inequality
$$\vert f(t)-f(s)\vert\leq \int_s^t{m(r) dr},$$
for $s<t$ and some $m\in L^1([0,T])$ then $f$ is absolutely ...
4
votes
4answers
326 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 ...
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1answer
77 views
Uniform approximation of increasing function in $C^{\infty}$
I have an increasing continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$ which is not differentiable everywhere, and I would like to approximate it with an infinitely differentiable function ...
4
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1answer
203 views
If $S\subset\mathbb R$ is a $G_\delta$ there is a function $\mathbb R\to\mathbb R$ continuous exactly on $S$. Reference?
Let $S\subset\mathbb R$ be a $G_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function ...
3
votes
2answers
211 views
A problem on chains of squares — can one find an easy combinatorial proof?
Consider the unit square $ S = [0,1] \times [0,1] $. For each $ n \in \mathbb{N} $, we can tessellate $ S $ by the collection
$$
A
= \left\{
\left[ \frac{i}{n},\frac{i + 1}{n} \right] \times
...
13
votes
1answer
343 views
Stromquist's 3 knives procedure
(copied from math.SE)
BACKGROUND: A cake has to be divided among 3 people with possibly different tastes, such that each person receives a single connected piece, and no person prefers another ...
3
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0answers
102 views
Is a continuous map between smoothable manifolds of the same dimension always smoothable?
(My question is inspired by this math.SE question, whose
negative answer I showed by a dimension-increasing map.)
Is it the case that for all smoothable manifolds $M_0$ and $M_1$ with the same ...
3
votes
3answers
315 views
Cardinality of $C^*([0,1])$ [closed]
What is the cardinality of the continuous dual of $C([0,1])$ (the set of continuous functions from $[0,1]\to \mathbb{R}$)?
3
votes
1answer
156 views
A functional equality
I don't know if this is known, but I was fiddling around with this equality :
$$f:(-1,1)\to (-1,1)\quad \text{satisfies}\quad(f(z)+1)^s=\sum_{j=0}^{\infty}\dbinom{s}{j}f(z^k)
\quad \forall z\in ...
1
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0answers
116 views
If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?
I have stumbled upon the following problem during my research:
Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...
2
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1answer
230 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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0answers
26 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 ...
2
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0answers
158 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 ...
7
votes
1answer
283 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 ...
8
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2answers
1k 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
229 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 ...
6
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3answers
859 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
147 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 ...
9
votes
3answers
1k 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 ...
0
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0answers
185 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 $z\in Q = \{ z : \Re z\in [a,b], \Im \sqrt{z} \in (0,1] \}$:
$$ ...
2
votes
0answers
60 views
Presentation of tree decompositions (and related concepts) in terms of continuous maps?
A tree decomposition of a graph $G$ is commonly defined in terms of a tree $T$ with the following structure:
Each vertex $t \in V(T)$ is associated to a set $X_t \subseteq V(G)$;
The union ...
0
votes
1answer
589 views
Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable?
Changing my question in light of Dan's answer. Thanks, Dan.
Consider a sequence of real random variables $X_i$ bounded in $L_1$, that is $\mathbb E\left|X_i\right|\leq M$ for all $i$. Suppose that ...
3
votes
2answers
136 views
continuity of length function $l: T(X) \times MF \to \mathbb R$
Let $T(X)$ be the Teichmuller space of a closed Riemann surface $X$ of genus $g \geq 2,$ and $MF$ the space of equivalence classes of measured foliations. Then we have a length function
$l: T(X) ...
3
votes
1answer
239 views
Automatic continuity of the inverse map
All topological spaces considered here are Hausdorff.
It is a well-known consequence of the minimality of a compact topology that an injective continuous map
$f\colon X\to Y$
where $X$ is compact, ...
0
votes
1answer
99 views
Continuity of critical points with respect to a parameterisation.
Hello.
I have a research note coming out soon, and I'm stuck showing that a weird kind of function is continuous. I need to it show a new method of bounding exponential growth factors in ...
2
votes
0answers
104 views
Is the Poincare action on the Klein-Gordon quantum field strongly continuous?
I am interested in checking continuity property of the Poincare group action on the Klein-Gordon quantum field theory defined over the Minkowski spacetime. Maybe the simplest example of QFT out there.
...
1
vote
2answers
571 views
Criteria for Lipschitz continuity
Is the following statement true. Assume that $f:[0,1]\to [0,1]$ is a continuous function such that $$\sup_t\lim\sup_{s\to t}\frac{|f(s)-f(t)|}{|t-s|}<\infty,$$ then $f$ is Lipchitz continuous.
2
votes
1answer
209 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 ...
3
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1answer
257 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
299 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
122 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
290 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.
7
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1answer
293 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 ...
0
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4answers
3k 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 ...
