# Questions tagged [continuity]

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129
questions

**3**

votes

**1**answer

109 views

### Functions with at most linear growth at infinity: is the constant itself continuous?

I am considering the family $\mathcal{F}$ of functions $f \colon \mathbb{R} \to \mathbb{R}$ which have at most linear growth at infinity, that is there exists a constant $M_f$ such that
\begin{...

**-4**

votes

**1**answer

84 views

### strict convexity and Lipschitz continuity [closed]

Consider a continuously differentiable function $f: \mathbb{R}^n \mapsto \mathbb{R}$. If $f$ is strictly convex, does it imply that it is not Lipschitz on $\mathbb{R}^n$?
Because if $f$ is strictly ...

**4**

votes

**1**answer

487 views

### Function whose sets of discontinuities and zeros are the rationals

Question: Is there a real valued function $f:\mathbb{R}\to\mathbb{R}$ such that its set of discontinuities is $\mathbb{Q}$ and its set of zeros $\{x\in \mathbb{R}\mid f(x)=0\}$ is also $\mathbb{Q}$?
...

**-1**

votes

**1**answer

101 views

### Sobolev injections [closed]

It is true to write that
$W^{1,\infty}(]0,\infty[) \hookrightarrow C([0,\infty[)$ et $W^{1,1}(]0,\infty[) \hookrightarrow C([0,\infty[)$ ?
Thanks

**0**

votes

**1**answer

154 views

### Is the pointwise supremum of a continuous function continuous?

Suppose $f(x , y)$ is continuous in both variables. For any $\epsilon > 0$ and some $y_0$, let $h_{\epsilon}(x) = \max_{y^{'}: \| y^{'} - y_0 \| \leq \epsilon} f(x , y^{'})$. It seems to me that $...

**3**

votes

**1**answer

121 views

### Is disintegration continuous?

Let $X,Y$ be Polish spaces and suppose that $X$ is compact. Denote by $\mathcal{Mes}(X,\mathcal{P}(X\times Y))$ the set of (Borel) measurable functions from $X$ to the set of Borel probability ...

**2**

votes

**1**answer

128 views

### Largest number N for which injective mapping $f: 2^N \to 2^8 \times 2^8 \times 2^8$ which is Lipschitz-1 CT with $K\leq 3$ exists

I have a function on $h: [0,1] \to [0,1]$ whose output is smooth (polynomial of low degree), and I need to discretize it but I need to save it with three 8 bit numbers. These three 8 bit numbers need ...

**1**

vote

**0**answers

20 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 ...

**-2**

votes

**1**answer

157 views

### Is a function piecewise continuous if it has a left-limit and a right-limit at every point in its interval domain and equals at least one of these? [closed]

Suppose a real-valued function f, whose domain is an interval, has the property that
at every point in its domain it has a left-limit and a right-limit, and it equals at least one of these. Is it ...

**-1**

votes

**1**answer

75 views

### If we restrict the domain of the $C^{1}$ function $f:[x_{1},x_{2}]\to[0,1]$ to an open or closed interval does it remain $C^{1}$? [closed]

I have a function $f:[x_{1},x_{2}]\to[0,1]$ which is $C^{1}$, $[x_{1},x_{2}]\subset[0,1]$ and $f$ surjective.
My first question is: can we always decompose the domain of $f$ into intervals $D_{k} = [...

**4**

votes

**1**answer

85 views

### Points of differentiability of squared distance from a point in metric spaces

Here the link to the same question I posted on MSE with no answer.
Let $(X,d)$ be a complete and separable metric space and let $I:=(0, + \infty)$. I recall the definition of absolutely continuous ...

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**0**answers

36 views

### Analyticity of $ \int\limits_x {\int\limits_y {Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}d{F_x}d{F_y}}}$

I need to show that the following function is analytic on the bounded complex plane. Lest define the function,
$f = \int\limits_x {\int\limits_y {Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x +...

**3**

votes

**1**answer

65 views

### Analyticity of $f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$ in the complex plane?

Let I have the following function,
$f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$
Where, $x,y \in C$, $a,b\in R$ and $- m \le \Re (x),\Re (y),\Im (x),\Im (y)...

**2**

votes

**0**answers

138 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_{...

**0**

votes

**2**answers

214 views

### Is the restriction map $C^1\ni f\mapsto\left.f\right|_K$ a continuous map?

Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $\iota$ be a continuous embedding of $\Theta$ into $C^1(E,E)$.
I would like to show that, ...

**1**

vote

**0**answers

155 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 ...

**5**

votes

**1**answer

194 views

### A functional equation in two complex variables

Let $X$ be a compact metric space, or just $X=\mathbb T$, the unit circle, if it helps. We consider only continuous, complex-valued functions on $X$.
Let $\varepsilon >0$. Is there $\delta > ...

**3**

votes

**2**answers

156 views

### When is the periodisation of a function continuous?

Consider a function $f\in\mathcal{C}_0(\mathbb{R})$, where $\mathcal{C}_0(\mathbb{R})$ denotes the space of bounded continuous functions vanishing at infinity. I am interested in the $T$-periodisation ...

**8**

votes

**0**answers

161 views

### 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 ...

**6**

votes

**1**answer

301 views

### A subcontinuous function, which is not continuous

Let $E$ be a Banach space and $T: E\rightarrow E$ be a mapping. $T$ is said to be subcontinuous if for any sequences $(u_n)_{n\in\mathbb{N}}$ in $E$ that converge strongly to $u$ the sequence $(T(u_n)...

**0**

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**0**answers

247 views

### Infinite composition of continuous functions

Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ be a sequence of functions and define $F_n:= f_n\circ \dots\circ f_1$. Then $F_n$ is continuous. However, the pointwise limit need not be (consider Mateusz'...

**1**

vote

**1**answer

138 views

### Baffling proof using function convexity

Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be convex, differentiable with derivative $f_x$ and Lipschitz continuous with constant $L$. Then, for $a,b,c,d \in \mathbb{R}$ such that $a \ge ...

**1**

vote

**1**answer

163 views

### Polynomial constraints on the values of continuous functions $\mathbb{R}\to\mathbb{R}$

Denote by $S$ the set of closed points in $X=\mathrm{Spec}\:\mathbb{R}[x_\alpha]$ ($\alpha \in \mathbb{Q}$) that have $\mathbb{R}$ as their residue field. There is an injective map from the set of ...

**2**

votes

**0**answers

69 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

50 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**

vote

**1**answer

157 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

152 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

38 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}\...

**5**

votes

**2**answers

198 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

72 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

117 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

196 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

78 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

117 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

624 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

254 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

108 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

329 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

191 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

61 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

103 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

116 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

104 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

70 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{...

**3**

votes

**1**answer

227 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

200 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

549 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

107 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

83 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

167 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 ...