# Questions tagged [continuity]

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107
questions

**2**

votes

**0**answers

56 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 $f$, ...

**0**

votes

**0**answers

31 views

### Differential operators and continuity

Let $Op(a)$ be a pseudo-differential operator, $Op(a) \colon S(\mathbb{R}^n) \to S(\mathbb{R}^n)$ is continuous.
My question: can we say by duality of $S$ and $S'$, that $Op(a) \colon S'(\mathbb{R}^n) ...

**-1**

votes

**0**answers

73 views

### Can we say that: $f(\beta)=\alpha$? [closed]

Let $X$ be a Hausdorff topological vector space and $f:X\to \mathbb{R}$ be an affine, sequentially continuous function.
Let $\{x_n\}\subset X $ be a sequence such that:
$$
\lim_{k}\frac{1}{k}\sum_{n=...

**1**

vote

**1**answer

143 views

### Continuity of $r\mapsto\int_{\Sigma\cap B_r(x)}f^2d\mu$

Let $\Sigma$ be an embedded smooth surface in $\mathbb{R}^3$, and let $f:\Sigma\to\mathbb{R}$ be a smooth function. Suppose $f$ is square-integrable on $\Sigma$, with
\begin{align}
0<\int_{\Sigma}f^...

**4**

votes

**1**answer

143 views

### Smoothness of the radius of convergence

Let $(x\mapsto a_n(x))_n$ be a sequence of smooth functions defined on some fixed interval $I$. Consider the power series $\sum_{n\geq 0}a_n(x)t^n$ and denote by $R(x)$ its radius of convergence. Does ...

**0**

votes

**0**answers

29 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}\...

**4**

votes

**2**answers

178 views

### Does convergence in law to absolutely continuous limit imply convergence in convex distance?

Let $(X_n)$ be a sequence of $\mathbb{R}^d$-valued random variables converging in distribution to some limiting random variable $X$ whose CDF is absolutely continuous with respect to the Lebesgue ...

**2**

votes

**0**answers

44 views

### 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 $A \subseteq [0, T]$ ...

**2**

votes

**1**answer

102 views

### Continuity of solution of a parabolic PDE w.r.t. system parameters

If we have a system of PDE of the form:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)=D\Delta y+F(t,x,f(x),y) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial \nu}(t,x)=0,\ (t,x)\in (...

**3**

votes

**1**answer

136 views

### Several definitions of Approximate continuity of real functions

I found the definition of approximate continuity stated as follows:
A function $f:\mathbb R \to \mathbb R$ is approximately continuous at $x_0$ iff there exists a set $A\in \mathcal L$ such that $x_0\...

**1**

vote

**0**answers

57 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 ...

**0**

votes

**1**answer

90 views

### Topologies and Borel $\sigma$-fields on disjoint unions

Consider a set of functions $\mathcal{F}$ on $E$ where $E \subset\mathbb{R}^k$ - e.g. the class of $L_1$ functions on $[0,1]$ - and endow it with a suitable metric $d$ that makes it Polish.
Consider ...

**2**

votes

**2**answers

250 views

### Supremum of continuous functions and essential supremum of continuous functions

Suppose that $(X,d)$ is a Polish metric space and $A$ is a set of continuous bounded functions $f:X\to \mathbb{R}$.
Suppose that $\mu:X\to[0,1]$ is a Borel probability measure.
Define
$$\sup A:X\to ...

**1**

vote

**1**answer

201 views

### Continuity of subharmonic functions

There is a result saying that the set where a subharmonic function defined on an open set of $\mathbb{R}^{m}$ ($m\geq2$) is discontinuous is a polar set. Could someone give me a reference for this ...

**1**

vote

**0**answers

91 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 , ...

**-2**

votes

**1**answer

281 views

### Injective uniformly continuous function $f:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$? [closed]

We say that a function $f: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$ is uniformly continuous if there is an integer $K\geq 1$ such that
whenever $(x,y),(x',y')\in \mathbb{Z}\times \mathbb{Z}$ with $|...

**5**

votes

**1**answer

186 views

### “Uniformly continuous” environment sum of a bijection $\varphi:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}$

Given any function $f: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$ we define the environment sum of $(x,y)\in\mathbb{Z}\times \mathbb{Z}$ with respect to $f$ by
$$\text{es}_f(x,y) = \sum\{f(x', y'): |(...

**1**

vote

**0**answers

60 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 ...

**2**

votes

**0**answers

96 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} = \...

**-2**

votes

**1**answer

104 views

### Continuity of the Restriction Map Between Function Spaces [closed]

Let $X,Y,Z$ be Hausdorff spaces and suppose that $Z\subset X$. Endow $C(X,Y)$ and $C(Z,Y)$ with the compact-open topologies and define the map $\rho$ as
\begin{align}
\rho:&C(X,Y)\rightarrow C(Z,...

**2**

votes

**1**answer

98 views

### Continuity of the derivations from semisimple Banach algebras

Let $A$ be a Banach algebra and $X$ a Banach $A$-bimodule. It is known that if $A$ is a $C^*$-algebra, then by Ringrose theorem every derivation $D:A\rightarrow X$ is continuous. Also, a famous ...

**1**

vote

**0**answers

69 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{...

**2**

votes

**1**answer

163 views

### Continuity of a parameterized convex optimization problem

I have a parameterised optimization problem:
\begin{align}
\boldsymbol{S}(p)= &\arg \min_{ \boldsymbol{x}} g( \boldsymbol{x})\\
\text{s.t. } & \boldsymbol{A}(p) \textbf{x} = \...

**2**

votes

**1**answer

184 views

### Can a bijection between function spaces be continuous if the space's domains are different?

It is well-known that any bijection $\mathbb{R} \rightarrow \mathbb{R}^2$ cannot be continuous. But suppose we have the two spaces $A = \{f(x):\mathbb{R^2}\rightarrow \mathbb{R} \}$ and $B = \{f(x):\...

**1**

vote

**1**answer

371 views

### Smallest Lipschitz Constant of a Differentiable Function [closed]

Let $X \subset \mathbb{R}^{n}$ be compact and convex. Moreover, let $f:X \rightarrow \mathbb{R}$ be a differentiable map with $\sup_{x \in X} \|\nabla f(x)\| = K < \infty$, where $\|\cdot\|$ ...

**1**

vote

**0**answers

94 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.

**5**

votes

**1**answer

82 views

### continuity of certain map which is defined on a Stonean space

Let $G$ be a discrete group which acts continuously on a Stonean space $\Omega$. Consider the map $f\colon \Omega\to \{0,1\}^G$ sending $x\in \Omega$ to $\chi_{G_x}$, where $\chi_{G_x}$ denotes the ...

**0**

votes

**1**answer

149 views

### An extension for lower semi continuous lower bounded real valued functions class

Let $(X,d)$ be a complete metric space. I need some explanations about the class of all functions like $f$ which have $f:X \to \mathbb{R}\cup\{ +\infty\}$, be a lower bounded and, for all $y \in X$ we ...

**4**

votes

**1**answer

147 views

### Continuity concepts for correspondences

Consider two metric spaces (X,d) and (Y,d') and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F ...

**1**

vote

**1**answer

146 views

### Is continuity preserved under norm operations

Let $F$ be a continuously differenable function over $\mathbb{R}$. Let $\Omega$ be
a bounded subset of $\mathbb{R}^2$. Assume that for every $w\in L^2(\Omega)$ then $v(x)=F(w(x))$, $x\in \Omega$, is ...

**3**

votes

**0**answers

219 views

### Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?

Recently I came to know about Atsuji space from the paper. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have found ...

**0**

votes

**1**answer

47 views

### Does $K^{1/2} (t,s)$ inherit the continuity of $K(t,s)$?

Assume that $K(t,s)$ is a (1) symmetric, (2) continuous, and (3) positive definite kernel on $[0,1] \times [0,1]$. The spectral decomposition of $K(t,s)$ is:
$$
K (t,s) = \sum_{i=1}^\infty \lambda_i \...

**5**

votes

**3**answers

359 views

### Continuity and sequential continuity of a linear functional

Let $E = C_c^0(\mathbb{R}^n;\mathbb{R}^m)$ be the space of compactly supported continuous functions on $\mathbb{R}^n$ with values on $\mathbb{R}^m$. There is a natural norm on this space: given $\...

**3**

votes

**0**answers

212 views

### Can continuity always be shown by using ε-δ? [closed]

When we learn calculus we usually:
1. Prove that polynomials, the exponential functions, the logarithmic functions, the trigonometric functions, the inverse trigonometric functions are continuous on ...

**0**

votes

**0**answers

62 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{\...

**2**

votes

**0**answers

214 views

### 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+\...

**5**

votes

**1**answer

308 views

### affine vs lipschitz

Let $(X,||\cdot||)$ be a normed space where $||\cdot||$ is the sup-norm and let $E$ be a convex and compact subset. Let $f:E\to [0,1]$ be continuous and affine, i.e. for all $x,y\in E$ and all $\...

**2**

votes

**0**answers

51 views

### $f: \Omega \rightarrow \mathbb{R}$ depending on finite coordinates is $\alpha$-Holder [closed]

Let $M = \{1, \dots, n\}$ be a metric space with the metric $d$ and, in $\Omega = M^{\mathbb{N}}$, define $\tilde{d}(x, y) = \sum_{k=1}^{+\infty} \frac{d(x_k, y_k)}{2^k}$.
We say that $f\colon \...

**10**

votes

**1**answer

390 views

### Does this Osgood-like condition imply continuity?

Let us consider a bounded, Borel function $F\colon \mathbb R^d \to \mathbb R^d$. Assume it satisfies the following
Osgood-like condition:
$$\tag{O}
\boxed{\vert \langle F(x) - F(y), x-y \rangle\...

**2**

votes

**2**answers

239 views

### Continuous monotone real functions of several real variables

Let $O$ be an open bounded connected set in $R^n$ and K its boundary. Given a continuous real function $f$ defined on $K$, I would like to extend $f$ to a continuous real function $g$ (i.e. $g$ ...

**1**

vote

**2**answers

178 views

### Can we get smooth parition of unity with uniformity?

Let $B \subseteq \mathbb{R}^n$ be a product of closed bounded intervals in $\mathbb{R}$. Fix $N>0$. Suppose I want to cover $B$ with $N$ open sets, $U_1, \ldots, U_N$, and get a smooth partition of ...

**0**

votes

**0**answers

143 views

### Continuity under various topologies for positive linear functionals

It is known that if $\mathcal A$ is a unital $\mathbb C$-$*$-algebra and $A$ is a unital subalgebra closed under $*$, and if $f : A \to \mathbb C$ is linear, then $f$ is positive if and only if $f$ is ...

**0**

votes

**1**answer

121 views

### Continuity of the solution of a Pde system

Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$ both continuous and bounded.
I have the following system of PDE's
\begin{align}
\begin{cases}
\frac{\partial}{\partial t} u_0(t,r)=- J* ...

**7**

votes

**2**answers

236 views

### $f$ locally (Lebesgue) integrable function on real line, $g(x):= \lim _{r\to \infty} \frac 1r \int_{x-r}^{x+r} f(t) dt$ exists for every real $x$

Let $f : \mathbb R \to \mathbb R$ be a function such that $f \in L^1[-a,a] , \forall a \in (0,\infty)$ and $g(x) : = \lim _{r\to \infty} \dfrac 1r \int_{x-r}^{x+r} f(t) dt$ exists in $\mathbb R$ for ...

**3**

votes

**0**answers

61 views

### Sub-quadratic Kolmogorov-Arnold?

The Kolmogorov-Arnold representation theorem says, essentially, that the only multivariate function you really need is addition. (Somewhat) more precisely, it says that for any continuous function $f:...

**1**

vote

**2**answers

257 views

### How to choose a continuous function which vanishes **only** on the closed set

We are reading John Roe's book Lectures on Coarse Geometry. We come across a question in P27 line 9:
Suppose $X$ is a paracompact and locally compact Hausdorff space, $\bar{X}$ is a ...

**4**

votes

**1**answer

175 views

### Induced maps on Hyperspace Topologies

If $X$ is a topological space let $2^X$ denote the set of closed subsets. There are multiple topologies one may equip $2^X$ with (in particular, I have in mind the Vietoris, Fell and similar ...

**4**

votes

**0**answers

110 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 ...

**1**

vote

**0**answers

121 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\...

**12**

votes

**2**answers

428 views

### Brouwer's Theorem in the free topos?

In Introduction to Higher-Order Categorical Logic, Lambek & Scott remark that Brouwer's Theorem (all functions $\mathbb{R}\to\mathbb{R}$ are continuous) holds in the free topos $\mathcal{T}$.
...