Questions tagged [continuity]

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Does there exist a continuous open map from the closed annulus to the closed disk?

(Originally from MSE, but crossposted here upon suggestion from the comments) In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
D.R.'s user avatar
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9 votes
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Does a generalization of Tietze's extension theorem hold for set-valued functions?

Let $X$ be a normal topological space. Tietze's extension theorem says that if $A \subset X$ is closed, then a continuous function $f: A \to \mathbb R^n$ can be extended to a continuous function whose ...
aduh's user avatar
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7 votes
0 answers
225 views

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 ...
Taras Banakh's user avatar
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5 votes
1 answer
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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 ...
Bremen000's user avatar
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5 votes
0 answers
303 views

Points of continuity of Kullback-Leibler divergence with respect to weak convergence

I know that the Kullback-Leibler $D(\mu||\nu) := - \int_K\log\big(\frac{d \nu}{d \mu}\big) \, d\mu,$ over probability measures on a compact $K$ subset of $\mathbb{R}^d$, is only weakly lower ...
thegain's user avatar
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5 votes
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375 views

Points of continuity of a lower semicontinuous function have non empty interior

Let $X\subset \mathbb R^d$ having non empty interior and let $f:X\to\mathbb{R}$ be lower semicontinuous. I know that the set of discontinuities of such a function is contained in a meager set, and ...
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4 votes
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Sub-quadratic Kolmogorov-Arnold?

The Kolmogorov-Arnold representation theorem says, essentially, that when computing a continuous function, the only multivariate function you really need is addition. (Somewhat) more precisely, it ...
Bill Bradley's user avatar
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4 votes
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112 views

Does there exist curve (for example, in $\mathbb R^2$) that either touches itself or intersects itself at every one of its points?

I really do not even know how to constructively think about this question that I wanted to post before, but delayed. I know that there are space-filling curves and curves of positive area and those ...
user avatar
3 votes
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168 views

Is the following generalization of piecewise continuity equivalent to any other common types of functions on metric spaces?

EDIT: I think what I have isn't precisely what I want... we should also require $x$ in condition (3) to be "not bad" in some sense, although I'm not quite sure what that should mean for my ...
exfret's user avatar
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Is there a generalization of the Kolmogorov-Chentsov continuity theorem for processes indexed by Banach spaces?

If $(X_t)_{t\ge0}$ is a real-valued stochastic process and for all $T>0$, there are $\alpha,\beta,C>0$ with $$\operatorname E\left[\left|X_s-X_t\right|^\alpha\right]\le C\left|s-t\right|^{1+\...
0xbadf00d's user avatar
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3 votes
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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 ...
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2 votes
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228 views

Functional continuity of eigenvalues?

We have the following theorems! Corollary VI.1.6 [Bhatia: Matrix Analysis]: Let $a_j(t)$, where $1\leq j \leq n$ be continuous complex valued functions defined on an interval $I$. Then there exists ...
VSP's user avatar
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Continuous analogue for Szpilrajn Theorem: complete preorder extends a continuous preorder

A corollary of Szpilrahn Theorem states: Any preorder on nonempty $X$ has a complete and transitive extension. I am thinking about the "Szpilrahn Theorem" for continuous preorder on ...
dodo's user avatar
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222 views

Convolution of continuous compactly supported functions on étale groupoid is continuous

Let $G$ be an étale Hausdorff groupoid, i.e. a topological groupoid $G$ such that the source and range maps $s,r: G \to G$ are local homeomorphisms. Consider the complex vector space $C_c(G)$ of ...
Andromeda's user avatar
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Is there a finite set of polynomials generating all rational numbers by 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. The ...
Ivan Meir's user avatar
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Are a.e. derivatives of continuous $VBG_*$ functions Denjoy–Perron integrable?

I would like to ask a question pertaining to the Denjoy–Perron (Henstock–Kurzweil) theory of integration. It is simple enough that I have entertained the idea that perhaps an answer is known, but I ...
David Manolis's user avatar
2 votes
0 answers
84 views

(Dis)continuity of periodic functions with non-summable Fourier series

Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$. We assume moreover that the square-summable Fourier coefficients of $f$, ...
Goulifet's user avatar
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Functions that are almost (left-) continuous almost everywhere

Denote the Lebesgue measure on $[0, T]$ as $\lambda(\cdot)$. Call a measurable function $f : [0, T] \to \mathbb{R}$ almost left-continuous almost everywhere if there exists an $A \subseteq [0, T]$ ...
RandomStudent's user avatar
2 votes
0 answers
226 views

Continuity of a constrained parameterized convex optimization problem

Consider the parameterized optimization problem: \begin{align} \boldsymbol{s}(p)= &\arg \min_{ \boldsymbol{x}} \quad g( \boldsymbol{x})\\ \text{s.t. } & \boldsymbol{A}(p) \textbf{x} = \...
Einar U's user avatar
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2 votes
0 answers
126 views

Left multiplication Fréchet Differentiable

Let $L^2$ be a separable Hilbert space and let $\{e_i(t)\}_{i \in \mathbb{N}}$ be a basis for it. Moreover let $\phi(T,z)$ and $\psi(T,z)$ be 1-parameter families (in z) in $L^2$, with basis ...
ABIM's user avatar
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94 views

On compactness in $C(X)$

Let $X$ be a Tychonoff space. It is well known, that for a family of scalar functions equicontinuity + pointwise boundedness imply relative compactness in $C(X)$ (with compact-open topology). It is ...
erz's user avatar
  • 5,385
2 votes
0 answers
576 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 ...
Rindra's user avatar
  • 21
2 votes
0 answers
84 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 $\...
Niel de Beaudrap's user avatar
2 votes
0 answers
155 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. ...
Yul Otani's user avatar
  • 342
1 vote
0 answers
78 views

Finding a functional that stays non-negative on a particular subset of $L^2[0, 1]$

Start with the Hilbert space $L^2([0, 1])$ with Lebesgue measure. Fix some Borel-to-Borel measurable function $f: [0, 1] \times [0, 1] \rightarrow \mathbb{R}$. I present 4 scenarios, each more ...
Daniel Goc's user avatar
1 vote
0 answers
21 views

Weakening compacity hypothesis in multifunctions intersection

Let $X,Y$ be metric spaces, $x^*\in X$ We define two multifunctions $F_1:X\rightrightarrows Y$,$F_2:X\rightrightarrows Y$. We recall the upper-semi-continuity in Berge's sense : A multifunction $F:X\...
Hamdiken's user avatar
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1 vote
0 answers
156 views

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 ...
user243245's user avatar
1 vote
0 answers
143 views

What can be said about cluster sets for power series of two variables?

I'm still trying to prove the continuity of a function $u$ which can be interpreted as the restriction of a power series of two variables, which I haven't managed to approach the right way yet. To ...
Raphael B's user avatar
1 vote
0 answers
65 views

Integrating over correspondences

Consider two compact sets $X$ and $Y$, a function $f:Y\to \mathbb{R}$, and a closed, non-empty correspondence $A:X\twoheadrightarrow Y$. Define the function $G:X\to \mathbb{R}$ by $$ G(x)=\int_{A(x)}f(...
tsm's user avatar
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1 vote
0 answers
35 views

How to prove continuity in topological group action of ${\rm{GA}}_a(X)$ on $T(X)$, to make ${\rm{GA}}(X)$ a topological group?

The question comes from the following paragraph of a text on geometry in the context of affine geometry (Marcel Berger et al., "Geometry I", P56-57): 2.7.1.3. If we don't want to resort to ...
zzzhhh's user avatar
  • 111
1 vote
0 answers
1k views

Weak sequential continuity vs strong continuity

Let $E$ be a Banach space, $T:E\rightarrow E$ a non-linear operator. $T$ is said to be Weakly Sequentially Continuous (shortly W.S.C) on $E,$ if for every $\left(x_{n}\right)_{n}\subseteq E$ with $x_{...
Motaka's user avatar
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1 vote
0 answers
728 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 ...
Motaka's user avatar
  • 291
1 vote
0 answers
125 views

On a continuous function as a substitute of the prime-counting function in the second Hardy–Littlewood conjecture satisfying certain asymptotics

It it well-known that the prime-counting function $\pi(x)$ satisfies the prime number theorem and that were in the literature two related conjectures to this arithmetic function, these are: the ...
user142929's user avatar
1 vote
0 answers
190 views

Maximum theorem with linear constraints. On parametric continuity of in optimization

Given \begin{align} s(\theta)= &\text{arg min}( g( \boldsymbol{x}) ) \\ \text{subject to }& \boldsymbol{A}(\theta) \boldsymbol{x} = \boldsymbol{b}(\theta) \\ &c_1 \le x_i \le c_2 , ...
Einar U's user avatar
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1 vote
0 answers
63 views

What are the various kinds of graphs that can be defined on $C(X)$

I was considering the space $C(X)$ where $X$ is a topological space and $C(X)$ is the set of all continuous functions from $X$ to $\Bbb R$. What are the various kinds of graphs that can be defined on ...
Learnmore's user avatar
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1 vote
0 answers
212 views

Norm closure of $C_b^1(\mathbb{R})$

I want to determine what the closure of $C_b^1(\mathbb{R})$, the space of continuous differentiable functions with bounded derivative, with respect to the supremums norm is. I think that $\overline{...
Gergana James's user avatar
1 vote
0 answers
246 views

Supremum of an almost surely continuous random process

I was learning this proposition and now I have a question, how to prove it for an almost surely continuous process? I would be very grateful for any tips.
Staysy's user avatar
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1 vote
0 answers
175 views

Continuity of a convex function on a vector bundle

Consider the rank-${n \choose m}$ vector bundle $\pi\colon E:=\bigwedge^m(TN)\to N$ over a smooth Finsler manifold $N$ and equip each fibre $E_q := \pi^{-1}(q)$ with a norm that depends smoothly on $q\...
Sven Pistre's user avatar
1 vote
0 answers
67 views

Extending an homotopy, knowing the two base functions extend

Let $A\subset B$ be paracompact spaces, and let $C$ be a paracompact space. Let $f_0,f_1:A\rightarrow C$ be continuous functions. $F:A\times[0,1]\rightarrow C$ a homotopy from $f_0$ to $f_1$. Suppose ...
Temitope.A's user avatar
1 vote
0 answers
373 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 ...
Jozsi's user avatar
  • 11
0 votes
0 answers
53 views

Reference request for equivalent Lipschitz smoothness conditions

For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
William Kong's user avatar
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0 answers
50 views

Why is von Neumann inequality important for equivalence of $\forall_j \ T_j^n\rightarrow 0$ in A-topology and abs continuity of $(T_1,\ldots, T_N)$?

The whole theorem goes as follows: Let $(T_1, \ldots, T_N)$ be a tuple of commuting operators in Hilbert space $H$ satisfying: $$\exists_{M > 0} \ : \ \forall_{p \in \mathbb{C}[z_1, \ldots, z_N]} \ ...
S-F's user avatar
  • 155
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0 answers
42 views

Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?

Below we use Bochner measurability and Bochner integral. Let $(Y, d)$ be a separable metric space, $\mathcal B$ Borel $\sigma$-algebra of $Y$, $\nu$ a $\sigma$-finite Borel measure on $Y$, $(Y, \...
Analyst's user avatar
  • 647
0 votes
0 answers
72 views

On "canonical" extensions of functions from integers to reals

Although this is essentially a port of my MathSE question, I think the users there tend to not understand how to interpret the questions from a higher perspective (and often too literally). This is ...
Graviton's user avatar
  • 109
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0 answers
162 views

Continuous dependence of the (infinite) roots of a polynomial on its coefficients

I'm trying to show the continuous dependence of the roots of a polynomial on its coefficients when the root number can be infinite (e.g., $x-y$). I don't know much about algebraic geometry but after I ...
Kryvtsov's user avatar
0 votes
0 answers
117 views

From convergence pointwise to convergence of the supremum for semicontinuous functions

Let $K\subset\mathbb{R}$ a compact set, and $(f_n)_{n\geq 1}$ and $f$ upper semicontinuous functions over $K$ (taking hence values in $\mathbb{R}\cup\{-\infty,+\infty\}$) such that for all $x\in K$, ...
G. Panel's user avatar
  • 557
0 votes
0 answers
81 views

Discontinuity of the Fourier transform of $ x \mapsto (1+ x^2)^{- \gamma/2}$ for $\gamma \leq 1$

Fix $\gamma > 0$. Let $\mathcal{F}$ be the Fourier transform and consider the function $f(x) = (1+ x^2)^{- \gamma/2}$ for $x \in \mathbb{R}$. This function is in $\mathcal{S}'(\mathbb{R})$ and its ...
Goulifet's user avatar
  • 2,174
0 votes
0 answers
58 views

Is $g = \sum_{n \in \mathbb{Z}} f(\cdot - n)$ continuous if $f$ is vanishing, continuous, and integrable?

Let $f \in \mathcal{C}_0(\mathbb{R}) \cap L^1(\mathbb{R})$ be a continuous and integrable function such that $f(x) \rightarrow 0$ when $|x|\rightarrow \infty$. The sequence of a functions $f_N = \sum_{...
Goulifet's user avatar
  • 2,174
0 votes
0 answers
68 views

Sufficient and necessary condition for the continuity of an improper integral

Let $f(\cdot) \in \mathscr{C}\left( \mathcal{D}; \mathbb{R} \right)$ where $\mathcal{D} \subseteq \mathbb{R}$ is open with $0 \in \mathcal{D}$ and $$ f(0) = 0, \quad \forall x \in \mathcal{D}\...
Johannes's user avatar
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0 answers
91 views

Continuity of solution to 2nd Order PDE w.r.t. the coefficients

I am considering the following 2nd order PDE : On a domain $R$, for some $r > 0$, \begin{equation*} \frac{1}{2}\sum_{i, j = 1}^{K}\gamma_{i}\gamma_{j}U_{x_{i}x_{j}}(x) + \sum_{i = 1}^{K}\frac{\...
Philo's user avatar
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