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Questions tagged [equicontinuity]

Let $(X, d_X)$ and $(Y, d_Y)$ be two metric spaces, and $\mathcal{F}$ a family of functions from $X$ to $Y$. The family $\mathcal{F}$ is equicontinuous at a point $x_0\in X$ if for every $\varepsilon > 0 $, there exists a $\delta > 0 $ such that $d_Y(ƒ(x_0),f(x) ) < \varepsilon$ for all $ƒ \in \mathcal{F}$ whenever $d_X (x_0, x) <\delta$.

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Tyrtyshnikov's proof that polynomial roots depend continuously on the coefficients

I'm having trouble understanding the following proof, which is taken from A Brief Introduction to Numerical Analysis by Eugene E. Tyrtyshnikov. (Note: The polynomials in the following are complex) ...
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3answers
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Is there a version of the Arzelà–Ascoli theorem capturing $C([0,\infty))$?

I only know the Arzelà–Ascoli theorem for continuous functions on a compact topological space. However, in the context of characterizing weak convergence of probability measures on $C([0,\infty))$, I'...
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33 views

is the following sequence equicontinuous

Let $f_k(x) = \frac{e^{-kx}}{k^2}$ where $x \in [0, 1]$, $k \in \mathbb{N}$. Is this sequence equicontinuous? What if $x \in \mathbb{R}$? The definition I have been given is: Let $F$ be a subset of $\...
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1answer
56 views

Is equicontinuity needed in theorem 7.25 part (b) from Rudin's “Principles”

About Rudin's theorem 7.25: 7.25     Theorem      If $K$ is compact, if $f_n \in \mathscr{C}(K)$ for $n=1,2,3,\dots,$ and if $\{f_n\}$ is pointwise ...
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1answer
51 views

Uniformly convergent subsequence of a uniformly bounded family of functions

I am trying to solve the following problem: Suppose $\{f_n\}$ is a sequence of functions that are continuous and differentiable on $[a,b]$. Suppose that the sequence $\{f_n\}$ and $\{f_n'\}$ are ...
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25 views

Sequence of solutions to a continuous ordinary differential equation in $\mathbb R^n$ has a convergent subsequence

How can I prove that a sequence of solutions to a continuous ordinary differential equation in $\mathbb R^n$ has a subsequence that converge to a limit ? I was thinking of using Arzela Ascoli theorem....
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Existence of Solution for Initial value Problem $y'(t)=\cos(ty), t>0$ and $y(0)=\frac{1}{n}, n\in\mathbb{N}$ using Ascoli Arzela

Consider the cauchy problem (1)\begin{cases} y'(t)=\frac{1}{1+ty}, & t>0 \\ y(0)=1+\frac{1}{n} & n\in\mathbb{N} \end{cases} (2)\begin{cases} y'(t)=\cos(ty), &...
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1answer
33 views

Show that a sequence admits converging subsequence

I don't know how to solve the following exercise. I think I should use Ascoli-Arzelà's theorem, but I don't know how. Let $\{ u_n\}_n$ be a sequence of functions in $C^1[0,1]$ with $u_n(0)=0$ for ...
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2answers
53 views

Compact operator by proving Ascoli-Arzelà

I need to prove that this operator satisfies Ascoli-Arzelà's hypothesis. $T: C^0[0,1] \rightarrow C^0[0,1] $, defined $Tu(x)=\int_0^x a(x,t) u(t)dt$, where $a(x,t)=C^0 ([0,1] \times [0,1])$. ...
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Show that this operator is not compact using Arzela-Ascoli

Let $T:C[0,1]\longrightarrow C[0,1]$ defined as $Tx(t) =tx(t)$. I need to prove that this operator is not compact using Arzela-Ascoli (using the Sup norm). I already prove that if X is a bounded ...
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1answer
31 views

Proof of equicontinuity via uniform convergence

I am studying for an entrance exam in August and I am trying to review and cover the real analysis section. I came across this problem on one of the past exams and I am having some trouble with it. ...
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1answer
70 views

Equicontinuous homeomorphism and compact metric

Let $(X, d) $ be a metric space and $f: X \rightarrow X$ be a homeomorphism on $X$. 1) We say that $f$ is semi-equicontinuous if for every $\epsilon > 0$ there exists $\delta > 0$ such that $...
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1answer
27 views

Sequence of smooth equicontinuous functions with unbounded derivatives

I found a very elegant construction in this link: https://math.stackexchange.com/a/311289/444015, I have no questions about this example. However, at least for me, it doesnt look like a natural ...
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1answer
53 views

continuous-sup property and equicontinuous set.

Suppose that $\Lambda \subset C^{0}$ is equicontinuous and bounded. (a) Prove that $\sup\{f(x)\mid f \in \Lambda\}$ is a continuous function. (b) Show that (a) fails without equicontinuity. ...
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The weaker boundedness implies uniformly bounded

Theorem (Arzelà-Ascoli). Eevery bounded equicontinuous sequence of functions in $C^{0}([a,b],\mathbb{R})$ has a uniformly convergent subsequence. The question asks to generalize the theorem with the ...
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1answer
27 views

Family of uniformly continuous functions, pointwise equicontinuous but is not uniformy equicontinuous.

I want to find a family of uniformly continuous functions $\{f_{n}\}$ such that $\{f_{n}\}$ is pointwise equicontinuous but is not uniformy equicontinuous. I'm having trouble finding an explicit ...
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1answer
11 views

Deriving an inequality for a set of functions that have equi-Lipschitz first derivatives.

let $ F =\{ F_t \}_{t \in \mathbb{N}}$ be a sequence of continuously differentiable real valued functions such that the sequence $ F' =\{ F'_t \}_{t \in \mathbb{N}}$ is equi-Lipschitz , i.e. there ...
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1answer
37 views

Does uniform continuity on a compact subset imply equicontinuity?

I gave a proof here some time ago but was wondering about the following: Suppose that $f_n:K\subseteq \Bbb{R}\to\Bbb{R}$ is continuous for each $n\in \Bbb{N}$, then $\{f_n\}$ is uniformly continuous. ...
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1answer
38 views

Sequence of continuous function converging pointwise to continuous function is equicontinuous?

I've proven the following "theorem": Let $I \subset \mathbb{R}$ be an interval, $(f_n: I \rightarrow \mathbb{R})_{n \in \mathbb{N}}$ be a family of continuous functions converging pointwise to a ...
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2answers
36 views

Analyse Equicontinuity (uniform continuity) of a function $ \frac{\sin(x)}{x},\; x > 0 $

How can I analyse equicontinuity of the function $$ \frac{\sin(x)}{x},\; x > 0? $$ When I draw the graph it is clear that this function is equicontinous, but I can not prove it analytically. Tried ...
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1answer
43 views

Convergence in Distribution implies Convergence in Expectation uniformly for an equicontinuous family

If $X_n \xrightarrow{\text{d}} X$, then for any equicontinuous family $\{f_\theta:\theta\in \Theta\}$, satisfying sup$\{|f_\theta(x)|:x\in \mathbf{R},\theta\in \Theta\}\}<\infty$, prove that $E[f_\...
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35 views

Equicontinuity of automorphism on compact subsets of the unit disk

Is the family of the functions $Aut(∆(1))$ equicontinuous on compact subsets of $ ∆(1)$? Here $∆(1)$ is the disk of radius $1$ centered at the origin on the complex plane. I am lost where to start. ...
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2answers
24 views

Pointwise equicontinuity

Is $\Phi:=\{(t \mapsto t^n):n \in \mathbb{N}\}\subseteq C([0,1),\mathbb{R})$ where $C([0,1),\mathbb{R})$ describes the set of all continuous functions from $[0,1) \to \mathbb{R}$ pointwise ...
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1answer
68 views

Equicontinuity of set of polynomials

Let $S = \{ p\in C([0,1]), p\ \text{polynomial of degree} \leq d: \max |p(x)| \leq 1 \}$ I want to show that $S$ is equicontinuous. Using the definition of equicontinuity we need for all $\epsilon &...
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1answer
33 views

Prove that $f_n(x)=\sin{\sqrt{x+4n^2\pi ^2}}\,,\;x\geq 0$ is equicontinous on $[0,+\infty)$

Prove that $f_n(x)=\sin{\sqrt{x+4n^2\pi ^2}}\,,\;x\geq 0$ is equicontinous on $[0,+\infty)$. converges pointwise to $0$ on $[0,+\infty)$. MY TRIAL \begin{align}\sqrt{x+4n^2\pi ^2}&=\sqrt{x+4n^...
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63 views

Prove that $\{f_n\}^{\infty}_{n=1}$ is equicontinuous $K$ if $\{f_n\}^{\infty}_{n=1}$ is uniformly convergent on $K$

Suppose that for each positive integer $n$, $f_n$ is a continuous function on $\Bbb{R}$ to $\Bbb{R}$. I want to show that if some subset $K$ of $\Bbb{R}$, the sequence $\{f_n\}^{\infty}_{n=1}$ is ...
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1answer
28 views

Monotonicity property of a sequence of equi-continuous functions

Suppose that a sequence of equi-continuous functions $\{f_n\}_{n \in \mathbb{N}}$ is such that for the fixed constant $K > 0$, each $f_n: [0, K] \to [0, K]$ and it is strictly increasing. Take ...
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1answer
48 views

A sequence of functions decreasing to 0 is equicontinuous in a compact metric space.

How to proof that? Let M be a compact metric space and $ \{f_n\} \subset C(M,\mathbb{R})$, so that $\{f_n\}$ is decreasing and $ lim f_n(x)=0 $, then $\{f_n\}$ is equicontinuous
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2answers
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If $K \subset \mathbb{R}^{n}$ is compact, then every equicontinuous set $\chi \subset C(K;\mathbb{R}^{m})$ is uniformly equicontinuous

Prove that if $K \subset \mathbb{R}^{n}$ is compact, then every equicontinuous set $X \subset C(K;\mathbb{R}^{m})$ is uniformly equicontinuous. I don't know any criteria to show that a given set is ...
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Convergent Subsequence of a Sequence in a Gelfand Triple

In my lecture notes of a class I took I found the following result. Unfortunately, I can't find a reference for the result. Does anyone have a reference or a hint where I could find the result? Lemma:...
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1answer
93 views

Example of an equicontinuous family of functions defined on a compact set that is not point-wise bounded.

As in the title, I am struggling to find a family of functions defined on a compact set that is both equicontinuous and not pointwise bounded. Any help is appreciated!
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1answer
95 views

Prove $\overline{E}$ (closure) is compact

In greater context, the question I'm trying to answer is: Suppose $E \subset C([0,1])$ is equicontinuous and assume there is some $x_0 \in [0,1]$ and constant $M$ s.t. $|f(x_0)| \leq M, \ \forall ...
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37 views

Equicontinuity or mean equicontinuity?

Can someone provide some examples to illustrate the difference between equicontinuity and mean equicontinuity? Can someone provide a concrete example that is mean equicontinuous but not ...
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1answer
28 views

Equicontinuity by bound in Lebesgue space

I need to apply the Arzela-Ascoli theorem for a sequence $f_n$. I already have the uniform bound and now it says $\frac{d}{dx} f_n(x)$ is bounded in $L^2([a,b])$ and I assume this yields the ...
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0answers
15 views

Equicontinuity with respect to a sequence of points? (terminology question)

I'm searching for an established term for "equicontinuity with respect to a sequence of points" for a function. Let $\{f_k\}$ be a family of functions and $\{x_k\}$ some associated points. My version ...
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2answers
79 views

Compact union of images of equicontinuous family

Suppose $\{f_n, n = 1, 2, \dots\}$ is an equicontinuous family of real-valued functions on a compact metric space $(X, d)$. What are some appropriate conditions ensuring that $\cup_{n = 1}^\infty f_n(...
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1answer
24 views

Equidifferentiability and Taylor series expansions

Suppose the sequence of vector valued functions $\{ {\bf f}_n \}$ are equidifferentiable at ${\bf x}_0$. In other words: $$\lim_{{\bf h} \to {\bf 0}} \max_n \frac{\left\Vert {\bf f}_n({\bf x}_0+{...
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2answers
294 views

Sequence $(f_n)$ equibounded, equicontinuous but with not uniformly convergent subsequence

Let $f_n:[0,\infty)\rightarrow R$, given by $f_n(x) = \sin(\sqrt{x+4\pi^2n^2})$. Prove that $(f_n)$ is equicontinuous and equibounded, but has no uniformly convergent subsequence. I'm having trouble ...
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2answers
216 views

Arzela-Ascoli theorem exercise

The question: Define a metric space $C(K)=\left \{ f: K\rightarrow \mathbb{R} > \right\} $ , where $f$ is continuous function on $K$. Let $K\in \mathbb{R}$ be compact and let $B\subset C(K)$...
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1answer
178 views

Equidifferentiable iff derivative is equicontinuous?

Let $\{f_n\}$ a sequence of function differentiable at $x_0$ We have equidifferentiability at $x_0$, if $\lim_{h \to 0} \max_n \left| \frac{f_n(x_0+h) - f(x_0)}{h}- f'_n(x_0)\right| = 0$ Are the ...
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1answer
50 views

Let $Y$ be a metric space and $H \subseteq C(X,Y)$ equicontinuous. Then the closure of $H$ is equicontinuous too.

Let $(X, \mathcal{T})$ be a topological space, $(Y,d)$ be a metric space, and $H \subseteq C(X,Y)$. Then, $\operatorname{cl}(H)$ is equicontinuous too in the pointwise topology (subspace on the ...
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1answer
124 views

Finite collection of continuous functions is equicontinuous.

Definition: Let $F \subseteq C(X,Y)$ where $X$ is a topological space, $(Y,d)$ a metric space. Then $F$ is called equicontinuous in $x$ iff $$\forall \epsilon > 0: \exists V \in \mathcal{V}(x): \...
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1answer
50 views

Equicontinuity of $ f^n (x) = \frac{1}{n} \cos(nx)$

I have to prove the equicontinuity of $ f^{(n)} (x) = \frac{1}{n} \cos(nx)$ and the unequicontinuity of $g^{(n)} (x) = \cos(nx)$ for $f^{(n)}, g^{(n)} := [0,1] -> \mathbb{R}$ I rarely heard ...
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1answer
73 views

Question about statements of Arzelà-Ascoli Theorem

I have some questions about the two versions of Arzelà-Ascoli Theorem given in Folland's book "Real Analysis" Version 1: Given a compact Hausdorff space $X$, let $F$ be an equicontinuous, ...
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1answer
243 views

Relationship between equicontinuity and total boundedness

A subspace of a complete metric space is compact iff it is closed and totally bounded. Take a compact space $X$, then its space of continuous functions $C(X)$ is complete. Subspaces of $C(X)$ are ...
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2answers
381 views

Supremum is continuous over equicontinuous family of functions

Problem: Let $\mathcal{M}$ be a metric space and let $\mathcal{F}$ be a bounded family of real valued functions on $\mathcal{M}$. Assume that $\mathcal{F}$ is equicontinuous. Define, for each $x \in \...
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1answer
54 views

Sequential characterization of non-equicontinuous sequence

Assume $f_n$ is a sequence of uniformly continuous functions. I was given the following criterion to determine if $f_n$ is equicontinuous. If $\exists t_n,s_n.t_n-s_n \to 0 \land \|f_n(t_n) - f_n(...
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1answer
41 views

Defining equicontinuity in terms of sequences

Let $X$ be a metric space and let $\{f_n\}$ be a sequence of uniformly continuous functions on $X$. Is it true that if I can show that for any convergent sequence $x_n$, $x_n \rightarrow x$, there ...
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2answers
200 views

How to show that functions are equicontinuous

Let $g\in A(\mathbb{D})$ (Dirichlet algebra), i.e., $g(z)$ is analytic in $\mathbb{D}=(|z|<1)$ and continuous on $\overline {\mathbb{D}}=(|z|\leq 1)$. How is it possible to show that: 1) The ...
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1answer
71 views

uniformly convergent implies uniformly equicontinuous?

We set $$C(\mathbb{R}) = \left\{ f : \mathbb{R} \to \mathbb{R}: f \text{ is continuous and } \sup_{x \in \mathbb{R}}\left|f(x) \right| < +\infty \right\}, $$ and assume that $(f_n)_{n\geq 1}$ is a ...