Questions tagged [continuity]

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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
1 vote
1 answer
106 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 ...
hmeng's user avatar
  • 129
4 votes
3 answers
414 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 ...
Isaac's user avatar
  • 2,727
4 votes
1 answer
141 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 ...
Simone256's user avatar
2 votes
1 answer
250 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 ...
muddy's user avatar
  • 69
15 votes
1 answer
748 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 ...
M. Winter's user avatar
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4 votes
1 answer
171 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\\ ...
Paul Joh's user avatar
  • 161
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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
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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
11 votes
2 answers
927 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 ...
Sonam Idowu's user avatar
2 votes
1 answer
162 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 ...
Bazin's user avatar
  • 15.1k
6 votes
1 answer
338 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→\...
Mikhail Katz's user avatar
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2 votes
0 answers
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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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
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 ...
user107952's user avatar
  • 2,063
18 votes
0 answers
1k 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 $...
D.R.'s user avatar
  • 741
2 votes
1 answer
350 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 \...
Kook Yeon-soo'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
  • 141
1 vote
1 answer
46 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 $m$-...
πr8's user avatar
  • 688
0 votes
1 answer
69 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, $x=a$, ...
dhp's user avatar
  • 11
3 votes
1 answer
158 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 $g:[ 0,1 ] ^ {n} \rightarrow [ 0,1 ] ^ {n}?$
Guill Guill's user avatar
5 votes
1 answer
231 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\...
UnclePetros's user avatar
7 votes
1 answer
491 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&...
jkjfgk's user avatar
  • 73
0 votes
2 answers
162 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 ...
dohmatob's user avatar
  • 6,716
0 votes
1 answer
155 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 $$ \mathbb{E}[(X_t - X_s)^2] \leq C~(t- s)^2, $$ ...
Drew Brady's user avatar
-1 votes
1 answer
123 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 ...
Grandes Jorasses's user avatar
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
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
0 votes
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
2 votes
1 answer
200 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 ...
Calvin Wooyoung Chin's user avatar
2 votes
1 answer
467 views

(Dis)prove : if every function with closed graph are continuous then the target space is compact

$(X, \tau_X) $ and $(Y, \tau_Y) $ be two topological spaces. $\forall f\in Y^X$ with $\text{Gr}(f) $ is closed implies $f\in C(X, Y) $. Question : Does this implies $(Y, \tau_Y) $ is compact? ...
Sourav Ghosh'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
3 votes
1 answer
302 views

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\...
Raphael B'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
2 votes
0 answers
38 views

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
  • 589
3 votes
1 answer
195 views

Maps that preserve winding numbers

This question is a cross post from the Math StackExchange since it got no attention at all there: https://math.stackexchange.com/questions/4414601/maps-that-preserve-winding-numbers I am looking for a ...
Manuel Eberl's user avatar
  • 1,181
3 votes
1 answer
624 views

Is a convex, lower semicontinuous function that is bounded from below, actually continuous?

While thinking about convex functions, I managed to put together the following proof which I find a bit too good to be true. $X$ is a topological vector space that is also a Baire space. Lemma: Let $f ...
iolo's user avatar
  • 611
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
2 votes
0 answers
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
  • 189
3 votes
2 answers
300 views

Smoothing a map $f:X\to \mathbb{R}$ while fixing it over a closed $C\subset X$

$\newcommand{\R}{\mathbb{R}}$I have a map $f\in C^0(X,\mathbb{R})$, where $X$ is a compact and Hausdorff topological space, which is a manifold outside of a compact subset $K\subset X$. I would like ...
Overflowian's user avatar
  • 2,523
5 votes
1 answer
309 views

Is $\mathbb{Q}$ the orbit of a continuous function that is computable when restricted to $\mathbb{Q}$?

In the previous post What is the smallest set of real continuous functions generating all rational numbers by iteration? I asked for the smallest set of continuous real functions that could generate $\...
Ivan Meir's user avatar
  • 4,782
1 vote
1 answer
102 views

Given an increasing function, need to construct a continuous increasing function equivalent to given function

Given an increasing function $f:[0,\infty)\to[0,\infty)$, we can define $$F(x)=\int_x^{x+1} f(t)dt,$$ which is continuous, increasing function satisfying $$f(x)\leq F(x)\leq f(x+1).$$ Question) For ...
Wilderness's user avatar
2 votes
0 answers
104 views

Can a smooth function always fit between two non-smooth functions? [closed]

Suppose I have two continuous functions, $f$ and $g$, with $f(x)<g(x)$ for all $x$ in some closed domain. Is it always possible to find a (piecewise, if needed) smooth function $h$ such that $f(x)&...
Sam Benner's user avatar
9 votes
2 answers
623 views

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 ...
Ivan Meir's user avatar
  • 4,782
2 votes
0 answers
73 views

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
  • 4,782
33 votes
2 answers
2k views

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 ...
Ivan Meir's user avatar
  • 4,782
2 votes
1 answer
180 views

Relationship between $C(X\times Y,Z)$ and $C(X,C(Y,Z))$

Let $X$, $Y$, and $Z$ be locally-compact, complete, and separable metric spaces and suppose that $X$ is compact; all non-empty. Consider the spaces $C(X,C(Y,Z))$ and $C(X\times Y,Z)$ both equipped ...
SetValued_Michael's user avatar
2 votes
0 answers
100 views

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
3 votes
0 answers
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
  • 479