Questions tagged [continuity]

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1 vote
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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 ...
4 votes
1 answer
750 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 paper1. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have ...
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)&...
9 votes
2 answers
622 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 ...
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 ...
2 votes
0 answers
72 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 ...
6 votes
3 answers
4k views

On the continuity of $\sum_{n=1}^{\infty} \sin(nx) / n^\alpha$

I know that the series $\sum_{n=1}^\infty \sin(nx) / n^\alpha$, with $0 < \alpha < 1$, converges for $x \in [0,2\pi]$. I'm trying to understand if the function $f(x) = \sum_{n=1}^\infty \sin(nx)...
6 votes
3 answers
2k views

Lipschitz continuity of singular values

How smooth are the singular values of a matrix $F$ in terms of entries of $F$? I am hoping for Lipschitz continuity, but was not able to find it.
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 ...
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 ...
0 votes
1 answer
291 views

Approximation of positive right-continuous function

Let $f:(0, +\infty)\to(0, +\infty) $ be a monotone decreasing, right-continuous function. Can we find a sequence $\{f_{n}\}_{n\in \mathbb{N}}$ of strictly monotone decreasing, continuous functions, ...
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 ...
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(...
2 votes
1 answer
2k views

Norms in Sobolev space $W^{1,\infty}$

Let $n\in\mathbb{N}$ and consider the Sobolev space $W^{1,\infty}(\mathbb{R}^n)=\lbrace u\in L^{\infty}(\mathbb{R}^n):\partial_iu\in L^{\infty}(\mathbb{R}^n) \rbrace$. A function is in $W^{1,\infty}$ ...
3 votes
1 answer
395 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{...
13 votes
2 answers
749 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 $\mathcal{T}$. ...
-4 votes
1 answer
266 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 ...
-1 votes
1 answer
113 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
3 votes
1 answer
547 views

upper semicontinuity in C(X)-algebras

I've stuck on a step of proposition 1.2 of Rieffel's article (continuous field of C*-algebras coming from group cocycles and actions, 1989). I think it basically proves that a C(X)-algebras coming ...
2 votes
1 answer
154 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 ...
0 votes
1 answer
817 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
286 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 ...
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 ...
-2 votes
1 answer
433 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 ...
4 votes
0 answers
98 views

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 ...
3 votes
1 answer
72 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)...
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_{...
0 votes
2 answers
308 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
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 ...
7 votes
1 answer
1k views

If $S\subset\mathbb R$ is a $G_\delta$, is there a function $\mathbb R\to\mathbb R$ continuous exactly on $S$?

Let $S\subset\mathbb R$ be a $G_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function $\...
5 votes
1 answer
213 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
556 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 ...
9 votes
0 answers
250 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 ...
5 votes
1 answer
371 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)...
1 vote
1 answer
170 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
168 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
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 $f$, ...
1 vote
1 answer
225 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^...
5 votes
1 answer
263 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
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}\...
5 votes
2 answers
371 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
1 answer
215 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 (...
2 votes
0 answers
167 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 $A \subseteq [0, T]$ ...
3 votes
1 answer
373 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
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 ...
1 vote
1 answer
486 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 ...
0 votes
1 answer
238 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
2k 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
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 , ...
3 votes
1 answer
686 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} = \...