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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21 views

Question on application of Lusin's Theorem

Suppose a function $U:T\times A\to \mathbb R$, where $T$ is a Polish space with finite Borel measure $\eta$ and $A$ is a compact set in a Polish space, is bounded, measurable and satisfies the ...
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39 views

Application of Arzela-Ascoli theorem: showing equicontinuity and pointwise boundedness

Let $\{f_n\} \subset C((0,1))$ be a sequence of functions such that $$\sup_{n \in \mathbb{N}}\{f_n(0) + f_n'(0)\} =:M<\infty$$ and there exists $\alpha \in (1, 2)$ such that $\forall n \in \mathbb{...
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Why is the image under integration of a uniformly bounded subset of $C[a,b]$ necessarily compact?

I am studying for qualifying exams and ran into this problem in Carothers: Define $T: C[a,b] \rightarrow C[a,b]$ by $(Tf)(x) = \int_a^x f$. Show that $T$ maps bounded sets into equicontinuous (and ...
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Prove the relative compactness of a sequence in $L^p$

Let $I=[0,1] \subseteq \mathbb{R}$ and let $(u_n),(v_n)$ be sequences in $C(I)$ such that $$|u_n(0)|+|v_n(0)| \le 1,~~~~ |u'_n(t)|+|v'_n(t)| \le t+e^t ~~~~ \forall t \in I, ~\forall n.$$ I would like ...
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Ascoli-Arzelà example and equicontinuity

My teacher did an example of Ascoli-Arzelà Theorem that I really don't understand, above all how he show that a series of functions is equicontinuous. Let $f_n:=x^n \quad , \quad x\in [0,1] \quad , \...
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27 views

Equicontinuity and “delta”

From the definition of a equicontinuous family, it's known that if $ \left |x-y \right |< \delta$ then $\left |f(x)-f(y) \right |< \varepsilon$ for all $f$ in the family. For all functions in ...
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Showing if a family of functions is equicontinuous, equibounded and/or equi-Lipschitz

I don't understand how to prove this kind of problem: Let $u_n:[0,1]\to \Bbb{R}$, $\qquad$ $u_n(x):=n+{\sqrt{\frac x n}}$ $\qquad \forall n \in \Bbb{N}$ Is $u_n$ equicontinuous, equibounded and/or ...
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$f_n$ equicontinous on unit circle implies $u_n$ equicontinuous on closed disk

I have the following question: Let $f_n$ are continuous real-valued functions on the unit circle and $u_n$ are harmonic functions on the open unit disk such and continuous on the closed unit disk. ...
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41 views

How to prove this closure is compact?

Let $(X, d)$ be a compact metric space, and $\mathscr{F}$ an equicontinuous family of functions from $X$ to itself. Suppose that $g: X → R$ is continuous. Show that the family $\mathscr{G} = \{g ◦ f: ...
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If $\{f_n\}$ is equicontinuous on $X$ then for all $x\in X$ there exists $f_x\in \mathbb{R}$ such that $f_n(x)\to f_x $

Let $X$ be a Banach space and $H$ be a countably dense subset of $X$. Let $\{f_n\}$ be a sequence of function $f_n:X\to \mathbb{R}$ such that $$ f_n(x)\to f_x\qquad \forall x\in H $$ We suppose that $\...
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140 views

Proving that a closed subset of $C[0,1]$ is compact.

Let $C=C[0,1]$ be the space of all continuous functions on $[0,1].$ $$K_n(a)=\{x.\in C:|x_0|\leq 2^n,|x_t-x_s|\leq N(a)|t-s|^a \enspace\forall |t-s|\leq 2^{-n}\},$$ where $N(a)=\frac{2^{2a+1}}{2^a-1},\...
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Every real and continuous function is the uniform limit of $a_0+a_1\cos(nx)\ldots$

I am studying the convergence of functions and equicontinuous families and I got the following problem: If we are working on the Interval $[0,\pi]$ I know that we can approximate a $f$ with Fourier ...
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45 views

Is the sequence of functions $f_n=\sin(nx)$ equicontinous in $\mathbb{R}$?

I've proved that the family $f_n=\sin(nx)$ is not equicontinous on the Interval $[0,1]$. We can prove also that the sequence is not equicontinous on any compact Interval. But I am wondering : Can we ...
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42 views

The family $f_n=\arctan(nx)$ is not equicontinous.

I am studying Equicontinuous families and the book says that $\arctan(nx)$ is not equicontinuous since the definition is violated if $x=0$ I would really appreciate if someone explain me what part of ...
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Prove that there exists a subsequence $\{g_{n_k}(x)\}$ converging uniformly to a continuous function $g(x)$ on $[0,1]$.

Let $f\in L^1[0,1]$, $E_n\subset[0,1]$ be measurable subsets and $$g_n(x)=\int_0^x\chi_{E_n}(t)f(t)\mathrm{d}t$$ where $\chi_{E_n}$ is the characteristic function of the set $E_n$. Prove that there ...
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Complex dynamics: does equicontinuity at a point imply equicontinuity in a neighborhood?

In Alan F. Beardon's book "Iteration of Rational Functions", the author defines the Fatou set $F(R)$ of a rational function $R$ as the maximal open subset of the Riemann Sphere (endowed with the ...
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34 views

Study the equicontinuity of a family of functions

Consider the family of functions $E=\{u_n:n \in \mathbb{N}\}$, where $u_n(x)=e^{-n||x||}$, $x \in \overline{B(0,1)} \subseteq \mathbb{R}^m$. I'm asked to study the equicontinuity of $E$ on the closed ...
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124 views

Proof for Uniform Convergence for $\{f_n\}$

Suppose $\{f_n\}$ is an equicontinuous sequence of functions defined on $[0,1]$ and $\{f_n(r)\}$ converges $∀r ∈ \mathbb{Q} \cap [0, 1]$. Prove that $\{f_n\}$ converges uniformly on $[0, 1]$. ...
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an example where $ \phi (x) = \sup_{n \in Z^+} f_n(x)$ is not continuous.

a) Give an example where $$ \phi (x) = \sup_{n \in Z^+} f_n(x)$$ is not continuous. b) Prove that if one assumes that the sequence {$f_n$} is an equicontinuous family, then $\phi : [0,1] \...
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What does it mean for an equicontinuous family of continuous functions to be uniformly equicontinuous?

Here is the definition I am given to equicontinuity: I found this question here: Is an equicontinuous family of uniformly continuous functions necessarily uniformly equicontinuous? but I do not ...
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Doubt about proof of equicontinuity

Let $X$ and $Y$ be two metric spaces and $B(X,Y)$ the metric space of the bounded functions from $X$ to $Y$. Then, the closure of an equicontinuous subset of $B(X,Y)$ is equicontinuous. I have ...
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61 views

Application of Arzela-Ascoli Theorem, proving uniform boundedness and equicontinuity

Consider the integral operator $\Lambda$ on $C([a, b])$ given by $(\Lambda f)(x) = \int_a^x f(t) dt. \ \ $ Prove that for any bounded set $S$ in $C([a, b])$, the set $\Lambda(S)$ is relatively compact ...
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28 views

$(f_n)$ be a uniformly equicontinuous and converges pointwise then it converges uniformly

Let $E$ be a compact metric space and let $(f_n)$ be a uniformly equicontinuous sequence of functions in $C(E)$. If $(f_n)$ converges pointwise on $E$, prove that it converges uniformly. $\textbf{My ...
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Prove the relative compactness of a family of $C^1$ functions

I'm solving the following exercise: Let $B = \{u \in C^1([0,1]): ||u'||_2 \leq 1\}$. Given a sequence $(u_n) \subseteq A = \{ u \in B : u(0)=0, u(1)=1\}$, show that there exists a subsequence ${u_n}...
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15 views

Equicontinuous Functions and Non-continuous functions uniformly bounded functions

I'm very confused on the idea of sequence of functions, I feel like it's very trivial and I'm overcomplicating it. For part a, I was thinking of constructing a family of functions ${f_n}$ such that ...
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27 views

Prove the equicontinuity of family of $C^1$ functions

I'm trying to solve the following exercise: Let $I=[0, 1] \subseteq \mathbb{R}$ and consider the family $$A= \{u \in C^1(I): ||u'||_{L^2} \leq 1 \}. $$ Prove that $A$ is a equicontinuous family. ...
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43 views

Equicontinuous on a compact set implies uniform equicontinuous

Theorem $(M,d), (N,\rho)$ metric spaces, $K \subset M$ compact. If $\mathcal{F} \subset F(M,N)$, where $F(M,N) = \{ f \mid f:M \rightarrow N\}$, is equicontinuous on $K$, then $\mathcal{F}$ is ...
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How equicontinuity follows in following proposition?

I was reading section 1.46 of Rudin functional analysis. He concluded E is equicontinous I do not know how? Please Help me Any Help will be appreciated
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Equicontinuity of set of functions.

The problem: Let $x(t) \in C[0,1]$. Then, define $x_n(t)=cos^nx(t)$ for all $n\in \mathbb{N}$. When is the set $(x_n)_{n=1}^{\infty}$ is equicontinuous in $C[0,1]$? Find a criterion in terms of the ...
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Let $E=[0,1]$, $K>0$ be a given constant, and $F$ be a set of continous functions on compact interval E such that $|f(x)-f(y)|\leq K|x-y|$

Let $E=[0,1]$, $K>0$ be a given constant, and $F$ be a set of continous functions on compact interval E such that $|f(x)-f(y)|\leq K|x-y|, x,y\in E, f\in F$ $\\$, and $f(0)=0$. Show that $F$ is ...
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35 views

Prove if exists $M>0$ such that $|f'(t)|\leq M\,\forall\,f\in F$ and for all $t\in(a,b)$ then $F$ is equicontinuous.

Let $F\subset C([a,b],\mathbb{R})$ such that all elements of $F$ are differentiable in $(a,b)$. Prove if exists $M>0$ such that $|f'(t)|\leq M\,\forall\,f\in F$ and for all $t\in(a,b)$ then $F$ is ...
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Show the Set of Polynomials of Degree $\le n$, Uniformly Bounded in a compact $C$ is Equicontinuous in $C$

Problem Show the Set of Polynomial $(p(x))$ of Degree $\le n$, Uniformly Bounded in a compact $C$ is Equicontinuous in $C$. I know what to do but I don't understand it very well. I've seen answers ...
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98 views

Show that $\{\sin(nx) \; \mid \; n \in N\}$ is not equicontinuous at $x=1$ [closed]

Show that $\{\sin(nx) \; \mid \; n \in N\}$ is not equicontinuous at $x=1$ I have not fully grasped the concept of equicontinuity and uniform equicontinuity. If someone can solve this properly it ...
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22 views

Show that $F=\{f_n\;|\; n\in N, \; f_n:[0,1]\to R \; \ni f_n(x)=x^n\}$ is not equicontinuous at $x=0$

Show that $F=\{f_n\;|\; n\in N, \; f_n:[0,1]\to R \; \ni f_n(x)=x^n\}$ is not equicontinuous at $x=0$ I am asked to prove it is not equicontinous at $x=0$ But I can't see how, if I take any $x$ ...
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Equicontinuity + Compact $\rightarrow$ Uniform eqcontinuity

My professor has defined equicontinuty and uniform equicontinuity to be two things: $H\subset C(X)$, $X$ metric space is called: Equicontinuous if: $\forall \epsilon>0\; \;y\in X, \exists \...
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225 views

prove/disprove: the functions $f_n(x)=\cos(x+n)+\ln\left(1+\frac{\sin^2(n^nx)}{\sqrt{n+2}}\right)$ are uniformly equicontinuous

Prove/Disprove that the sequence of functions $(f_n)$ from $\Bbb R$ to $\Bbb R$ defined by $$f_n(x)=\cos(x+n)+\ln\left(1+\frac{\sin^2(n^nx)}{\sqrt{n+2}}\right)$$ are uniformly equicontinuous. That ...
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Could someone elaborate on the difference between different definitions of stochastic equicontinuity (Asymptotically uniformly equicontinuous)?)

Here is the question, I am quite new to the field of Asymptotic Statistics and I'm kind of confused by the different definitions of stochastic equicontinuity in various resources. In particular, the ...
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Two points at proof of Ascoli Arzela Theorem

enter link description here I was working on the proof of Ascoli-Arzela Theorem (10.3) at the link above. However two points in there are not clear for me. 1st For necessity, we take $\mathcal F$ as ...
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Is equicontinuity independent of the chosen metric of the state space

There already exist several posts on similar questions, but I could not find any, which really gives a precise answer to my problem. Let $A \subseteq C([0,T],E)$, where $E$ is a topological space and ...
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110 views

supremum of uniform bounded equicontinuous functions is continuous

I am working on a problem in Berkeley book but it has no solution. I want someone give me comments on my proof. Here is the problem : Let $\mathcal{F}$ be a uniformly bounded, equicontinuous family ...
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96 views

Peano's Existence Theorem: Questions about the proof

Since there are already quite a few questions about the Peano Existence Theorem out there, I would still like to post the version from my textbook here inorder to grasp it's steps. Let $G\subset\...
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Determine family, $\mathcal{F}$ = sequence of $f_{n}(x)$ that are $C_{n}$ - Lipschitz and $C_{n} \to C$ in $[0,1]$ is compact - Proof Verification

Determine if the family, $\mathcal{F}$ = sequence of $f_{n}(x)$ that are $C_{n}$ - Lipschitz and $C_{n} \to C$ in $[0,1]$ is compact. To determine if this set of functions is compact we have to ...
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Does this “local” version of equicontinuity have a name?

Consider a sequence of functions $\{ f_n\}_{n \geq 1}$, where each $f_n : X \to \mathbb{R}$ for a metric space $X$. Let $r_n$ be a sequence of positive reals diverging to $+\infty$, and suppose that ...
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1answer
100 views

Arzelà- Ascoli Theorem and polynomials

I am studying Real Analysis and I need to proove this exercise: Let $(Pn)$ a sequences of polynomials, with degree $\leq m$. If $(Pn)$ is uniformly bounded on a compact set $K$ then there is a ...
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40 views

Show that $(Tf_{n})_{n}$ is equicontinuous

Let $(f_{n})_{n}\subset C([0,1])$ such that $\vert\vert f_{n} \vert \vert_{\infty}\leq1$ for all $n \in \mathbb N$. Furthermore let $k \in C([0,1]^{2})$ define $T: (C([0,1]),\vert\vert \cdot \vert \...
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242 views

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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313 views

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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42 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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119 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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317 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 ...