Questions tagged [continuity]
The continuity tag has no summary.
203 questions
0 votes
0 answers
51 views
Pointwise supremum representation of bounded functions on a strengthened topology
Let $(X, \tau)$ be a topological space and let $\varphi \colon X \to \mathbb{R}$ be a function. We define $\tau_\varphi$ as the smallest topology containing $\tau$ such that $\varphi$ is continuous. A ...
- 291
6 votes
1 answer
203 views
On the uniqueness of the extension of the Dirac delta measure from weakly to norm-continuous functions
Let $H$ be a separable infinite-dimensional Hilbert space over $\mathbb{R}$ with inner product $\langle \cdot, \cdot \rangle$ and norm $|\cdot|$. Define the function $f\colon H \to \mathbb{R}$ by $$ f(...
- 291
6 votes
2 answers
724 views
Approximate a continuous function by uniformly continuous ones
Let $(X, d)$ be a complete and separable metric space. I am interested in the case where bounded subsets of $X$ are not necessarily compact. Let $f: X \to \mathbb R$ be bounded and continuous. Is ...
- 1,163
4 votes
1 answer
205 views
An existence problem of stopping time with respect to continuous stochastic process
Let $Y$ be a continuous stochastic process on $[0,T]$ with a complete filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ satisfying the usual condition. Let $\tau$ be a ...
- 41
23 votes
5 answers
3k views
Are eigenspaces continuous?
There is a class of results of the form "eigenvalues are continuous in square matrices" (e.g., this MSE question and its answers). An analogous question is whether the eigenspaces of an $n \...
- 3,255
7 votes
1 answer
521 views
A particular continuous selection problem
Let $B$ and $S:=\partial B$ denote, respectively, the unit ball and the unit sphere wrt to some norm $\|\cdot\|$ on $\Bbb R^2$. For a fixed real $r\in(0,1]$ and all $u\in S$, let $$D_u:=(B+ru)\cap(B-...
- 143k
5 votes
1 answer
302 views
A continuous selection problem
Let $(B_t)_{t\ge0}$ be a continuous (say wrt the Hausdorff metric) family of nonempty centrally symmetric convex compact subsets of $\Bbb R^2$. For a norm $\|\cdot\|$ on $\Bbb R^2$ and each real $t\...
- 143k
41 votes
2 answers
1k views
Continuity-minimal topologies
Motivation. Let $X$ be a non-empty set. If $\tau$ is the trivial topology $\{\varnothing, X\}$ or the discrete topology ${\cal P}(X)$, then every function $f:X\to X$ is continuous. For the topologies ...
- 53.5k
5 votes
1 answer
396 views
Finding a special non trivial topology on the unit interval
I am seeking a non-trivial topology on the unit interval $[0, 1]$ (neither discrete nor indiscrete) such that the following four functions $f$, $g$, $h$, and $t$, defined from $[0, 1] \times [0, 1]$ ...
- 59
5 votes
0 answers
152 views
Is the Nemytskii map $u \mapsto \max(0,u)$ Hölder continuous from $H^1_0(\Omega)$ into $H^1_0(\Omega)$?
I recently read that there is a chance that the positive part function $f(x) := \max(0,x)$ is such that the associated Nemytskii map $f\colon H^1_0(\Omega) \to H^1_0(\Omega)$ is $1/2$-Hölder ...
- 177
26 votes
2 answers
994 views
Does every closed and infinite-dimensional subspace of $C[0,1]$ contain a non-zero function with uncountable zero set?
As stated in the title, does every closed and infinite-dimensional subspace of $C[0,1]$, the space of continuous functions on the unit interval, contain a non-zero function whose zero set is ...
- 11.7k
1 vote
1 answer
147 views
Continuity of Solutions to Linear Programs
Suppose I have a linear program of the form $\max c^Tx$ such that $Ax \leq b$. I am curious as to the continuity of the solutions to such a linear program under perturbations of the entries of $A$. ...
- 143
9 votes
1 answer
436 views
Continuous $f$ so that $f(x)$, $f(x) + \sqrt{2}$, $f(x) + x$ $\in \mathbb{Q}^c$ when $x \in \mathbb{Q}^c$
Does there exist a continuous function $ f:\mathbb{R} \to \mathbb{R} $ such that $$ f(x),f(x) + \sqrt{2} , f(x) + x $$ are in $\mathbb{Q}^c$ for all $ x \in \mathbb{Q}^c $? Here, $ \mathbb{Q}^c $ ...
- 1,211
2 votes
0 answers
85 views
Construct a differentiable function whose gradient has a prescribed modulus of continuity
$\newcommand{\bR}{\mathbb{R}}$ Let $\alpha := e^{-(1 + \sqrt{2})}$. We define the following modulus $\psi : \bR_+ \to \bR_+$ of continuity $$ \psi (x) := \begin{cases} 0 &\text{if} \quad x =0 , \\ ...
- 1,163
7 votes
2 answers
276 views
Hölder continuity of Green function for simply connected domains
I read in a paper by E. A. Rakhmanov "Orthogonal Polynomials and S-curves" the following statement, in Lemma 9.9 therein, which I state here in my words. Under the standard hypotheses for ...
user528012
1 vote
0 answers
82 views
Regularity of homogeneous random field in $\mathbb R^2$ and absolute continuity. Has this been generalized to the $n$-dimensional case?
What I am talking about is a rather old mathematical paper, published in Russian, from 1957. The name of the paper is “О линейном экстраполировании дискретного однородного случайного пoля”. I cannot ...
- 53
0 votes
1 answer
217 views
Counterexample wanted: Banach space but not BK-space
What is an example of a Banach space that is not a BK-space? A normed sequence space $X$ (with projections $p_n$) is a BK Space if $X$ is Banach space and for all natural numbers $n$, $p_n(\bar{x}) = ...
- 1
7 votes
2 answers
399 views
For this continuous non differentiable function $f$ How to determine $\sup\{a\}$ s.t $\lim\limits_{h\to0}\frac{f(x+h)-f(x)}{h^\alpha}=0$ for all $x$?
I asked this question on MSE here. Define $g(x)= |x|$ for $|x|\in [-1,1]$ , $g(x+2)=g(x)$ $$f(x)= \sum_{n \ge 1} \frac{3^n g\left(4^n x\right) }{4^n}$$ This function is a famous example of a ...
- 697
2 votes
1 answer
119 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 votes
1 answer
158 views
Is this function on the Cantor set continuous? [closed]
Let $S = \displaystyle \prod_{n \ge 1} \{ 0, 1\}$ be the set of binary sequences, so $S$ with the product topology is homeomorphic to the Cantor set. Endow $\mathbb{Z}_{\ge 1} \cup \{ \infty \}$ with ...
- 233
-1 votes
1 answer
155 views
Lipschitz function which is surjective on subset implies that the subset is dense
Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a Lipschitz-function. Suppose $A \subseteq \mathbb{R}^n$ is an $(n-1)$-connected subset such that $f(A) = \mathbb{R}^n$. I would like to show that $A\subseteq ...
- 684
1 vote
0 answers
215 views
Sum of upper semi continuous and lower semi continuous functions
Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
- 1,045
1 vote
1 answer
174 views
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
4 votes
3 answers
610 views
Does the uniform boundedness principle holds for multilinear maps as well?
This question has been motivated by weak* completeness of distributions. According to the answer in the above post, any barrelled locally convex topological vector space $E$ satisfies the uniform ...
- 3,745
4 votes
1 answer
289 views
Solution of SDE at finite time, continuity of pdf
I'm looking at the Langevin dynamics described by the following SDE $$d X_t = - \nabla U(X_t) \, d t + \sqrt {2 \Sigma} \, d B_t,$$ where $X_t \in \mathbb R^d$, $\nabla U(\cdot)$ has some regularity ...
- 69
2 votes
1 answer
339 views
Continuity of eigenvector of zero eigenvalue
Wonder whether anyone has an idea on showing the following or to point out that it is not true: Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
- 69
16 votes
1 answer
874 views
What is this equivalence relation on topological spaces: there are bijective continuous maps in both directions
Consider the following equivalence relation on topological spaces: $X\sim Y$ $:\Longleftrightarrow$ there are bijective continuous maps $\phi:X\to Y$ and $\psi:Y\to X$. Note that there are no ...
- 14.5k
4 votes
1 answer
219 views
Reference request: Solution to second order parabolic linear BVP belongs to $\mathcal{C}(0,T;H^1(\Omega))$
I am currently reading the paper [1]. In Theorem 3.1. b) the following boundary-value problem is given: \begin{align*} \partial_{t} y - \Delta y + g\cdot y = f \text{ in } ]0,T[ \times \Omega\\ ...
- 161
1 vote
0 answers
116 views
Finding a functional that stays non-negative on a particular subset of $L^2[0, 1]$
Start with the Hilbert space $L^2([0, 1])$ with Lebesgue measure. Fix some Borel-to-Borel measurable function $f: [0, 1] \times [0, 1] \rightarrow \mathbb{R}$. I present 4 scenarios, each more ...
- 93
11 votes
2 answers
1k views
Twice continuously differentiable implied by existence of limit
I have the following question. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions (vanishing at infinity) and assume that $$ \frac{f(x+t)+f(x-t)-2f(x)}{t^2}\to g(x) $$ for all $x\in X$ when ...
- 129
2 votes
1 answer
277 views
Hausdorff dimension of the curve of a continuous nowhere differentiable function
It is of course well-known that there are plenty of functions from $\mathbb R$ into itself which are continuous and nowhere differentiable. Although the Baire Category Theorem is enough to prove the ...
- 16.6k
6 votes
1 answer
469 views
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→\...
- 18k
2 votes
0 answers
558 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 ...
- 267
0 votes
0 answers
70 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, \...
- 657
34 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 ...
- 2,183
18 votes
0 answers
2k views
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 $...
- 1,235
2 votes
1 answer
567 views
A continuous injection from $[0,1]$ to $\mathbb{R}^2$
Consider the continuous and injective mapping \begin{eqnarray*} \varphi:[0,1] &\rightarrow& \mathbb{R}^2, \\ t &\mapsto& (x(t),y(t)), \end{eqnarray*} such that $x(0)<x(1)$, and \...
- 147
1 vote
0 answers
38 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\...
- 141
1 vote
1 answer
92 views
From relative convexity to modulus of continuity estimates for the dual gradient mapping
Let $F: \mathbf{R}^d \to \mathbf{R}$ be a convex function, let $m > 0$, and define $Q_m: \mathbf{R}^d \to \mathbf{R}$ to be the mapping $x \mapsto \frac{m}{2} \| x \|_2^2$. One says that $F$ is $m$-...
- 892
0 votes
1 answer
134 views
Convergence in expectation of a discontinuous function
Consider a random variable $X\in \mathbb{R}^d$. Let ${\theta_m}$ be a sequence of real numbers that converge to $\theta$. Let $f(x,y)$ be a function that is not continuous. To be specific, fix, $x=a$, ...
- 11
3 votes
1 answer
240 views
Homeomorphic extension of a discrete function
Let $f : \{ 0,1 \} ^ {n} \rightarrow \{ 0,1 \} ^ {n}$ be a bijective map. Then is there a known computable way to extend it to a homeomorphism $g:[ 0,1 ] ^ {n} \rightarrow [ 0,1 ] ^ {n}?$
- 31
5 votes
1 answer
302 views
On the continuity of a Set-Valued function (correspondence) [closed]
Let $f:\mathbb{R}^{n}\rightrightarrows \mathbb{R}^{m}$ be a set-valued function defined by \begin{equation*} f\left( x\right) =\left\{ y\in \mathbb{R}^{m}:g\left( x\right) +h\left( x\right) ^{T}y\...
- 61
7 votes
1 answer
632 views
Non-homeomorphic connected one-dimensional Hausdorff spaces that have continuous bijections between them in both sides
I need to construct an example of two non-homeomorphic connected one-dimensional Hausdorff spaces that have continuous bijections between them in both sides. Spaces should have induced ("good&...
- 73
0 votes
2 answers
210 views
Asymptotics of the unique root of a polynomial equation defined as a sum of rational expressions
Let $\lambda_1\ge \ldots \ge \lambda_n \gt 0$. Define a function $F:\mathbb R_+ \to \mathbb R_+$ by $$ F(t) = t^2\sum_{i=1}^n\frac{\lambda_i^2}{(\lambda_i + t)^2}. $$ It is clear that $F$ is strictly ...
- 7,033
0 votes
1 answer
314 views
Constructing a Gaussian process on $[0, 1]$ such that the sample paths are $1$-Lipschitz continuous with high probability?
In the paper [1] the authors demonstrate that for a centered Gaussian process $\{X_t\}_{t \in [0, 1]}$, if there is a constant $C > 0$ such that $$ \mathbb{E}[(X_t - X_s)^2] \leq C~(t- s)^2, $$ ...
- 542
0 votes
1 answer
234 views
Under which conditions Mean Square Continuity implies Sample Continuity for Gaussian Processes?
First, let us give the setting. Let $(\Omega, \Sigma, \mathbf{P})$ be a probability space, let $T$ be some interval of time, and let $X: T \times \Omega \rightarrow S$ be a stochastic process. By Mean ...
- 141
1 vote
0 answers
186 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 ...
- 11
0 votes
0 answers
140 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 ...
- 109
0 votes
0 answers
260 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 ...
- 9
2 votes
1 answer
515 views
A characterization of continuity in terms of preservation of connected sets. Where to find the result?
There is a result that if $X$ is a locally connected space and $Y$ is a locally compact Hausdorff space, then a function $f \colon X \to Y$ is continuous if and only if $f$ has a closed graph and for ...
