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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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56 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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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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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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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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107 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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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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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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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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60 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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109 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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203 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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53 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
38 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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181 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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58 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 ...
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86 views

Characterization of Compactness in $\ell^\infty$

The metric space $\ell^\infty$ consists of all bounded real valued sequences with the metric is $ d(x,y) = \sup_{i \in \mathbb{N}} | \xi_i - \eta_i | $, where $x = (\xi_i), y = (\eta_i)$. I have been ...
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163 views

Is an equicontinuous family of uniformly continuous functions necessarily uniformly equicontinuous?

Let $X,Y$ be metric spaces and $(f_n)_n$ a family of functions $X \rightarrow Y$. We say that $(f_n)_n$ is equicontinuous if $\forall x\in X \quad \forall \varepsilon >0 \quad \exists \delta >0 ...
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Equicontinuity of a family which is a sequence

If we consider a family of functions to be a sequence of functions, do we need to require in the definition that $$|f_n(x)-f_n(y)|<\epsilon$$ holds for $n$ large? Or should it hold for all $n$? ...
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Show that $A$ is both closed and open

Let $X$ be compact, $\mathcal{F}$ be a family of equicontinuous real valued functions on $X$. Define $A:=\{a\in X|\{f(a): f\in\mathcal{F}\}\text{ is bounded}\}$. Show that $A$ is both closed and open. ...
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105 views

A corollary of Arzela-Ascoli

The Arzela-Ascoli Theorem says that If $X$ is a compact metric space and $F$ a subset of $C(X)$, then $F$ is compact if and only if $F$ is closed, uniformly bounded, and equicontinuous. My ...
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62 views

Equicontinuity of family of polynomial approximations

Suppose $f:[0,1]\to \mathbb{R}$ is uniformly continuous, and $(p_n)_{n\in\mathbb{N}}$ is a sequence of polynomial functions converging uniformly to $f$. Does it follow that $\mathcal{F}=\{p_n\mid n\...
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185 views

Show that any compact subset of $C([0,1])$ with sup norm is equicontinuous [closed]

Question is stated as follows: Show that any compact subset of $C([0,1])$ with the sup norm $\lVert \cdot\rVert_{\infty}$ is equicontinuous.
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258 views

Is the set of derivatives of analytic functions of the unit disk equicontinuous?

Let $F$ be the set of analytic functions from the (open) unit disk $\mathbb{D} $ to itself. Let $G$ be the set of derivative functions of $F$. I'm trying to show that $G$ is not an equicontinuous ...
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68 views

Proving compactness of a subset

Let $I = [a, b]$ be an interval in $\mathbb{R}$. We consider $C(I)$ with the maximum norm and define a subset of $C(I)$ $$M_c=\left\{f \in C^1(I): \int_a^b\vert f(x) \vert ^2dx + \int_a^b\vert f'(x) ...
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180 views

A Family of functions that is closed but not bounded.

I have to show that the following family of functions: \begin{align*} \mathfrak{F}= \left\{f \in C\left([-1,1]\right) : \int_{-1}^{1}f(x)dx \in [0,1]\right\} \end{align*} is closed; but that it is ...
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193 views

Example 7.21 and Definition 7.22 in Baby Rudin: Why is this sequence of functions not equicontinuous?

Here is Definition 7.19 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\left\{ f_n \right\}$ be a sequence of functions defined on a set $E$. We say that $...