# 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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### Understanding the proof of a simplfied version of the Aubin-Lions lemma

I am following the textbook The Three-Dimensional Navier-Stokes Equations by Robinson, Rodrigo, and Sadowski. I am confused on some parts of their proof for a simplified version of the Aubin-Lions ...
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### If $G\in C([0,1])$ and strictly increasing, can we find a sequence $G_n\n C^{\infty}$ with uniformly equicontinuous density?

Let $G:[0,1]\rightarrow [0,1]$ be a strictly increasing and continuous cdf with $G(1)=1$. I have proven some property for $G\in C^{\infty}([0,1])$ that relies on the continuity of $g(x)=G'(x)$. I hope ...
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### Understanding equicontinuity in asymptotic normality with nonsmooth objective functions

I am working on the normality of extremum estimators with nonsmooth objective functions. Assume that my objective function is $Q_n(\theta)$, where $n$ is the sample size. I denote by $\hat \theta_n$ ...
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### Equicontinuity of a set

this is an excercise in an old test from my course of analysis. Let $\rho : \mathbb{R} \to [0,\infty]$ continuos such that $\int_{\mathbb{R}}\rho(t)dt=1$. Let $f_n$ secuence of real functions, bounded ...
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### a sequence of characteristic functions converge uniformly near t=0, then they are equicontinuous

I encountered a problem when reading A course in probability theory by Kailai Chung: If the sequence of ch.f.'s $\{f_n\}$ converges uniformly in a neighborhood of the origin, then $\{f_n\}$ is ...
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### Prove that $X$ is relatively compact in the space of continuous functions on $[-1,1]$

Let $X$ be the set of real-valued functions $g:\mathbb{R} \to\mathbb{R}$ of class $C^2$, with support in $[-1,1]$, and such that $\int_{-1}^1(g'')^2\le1$. I am interested in proving that the closure ...
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### Equicontinuity is not an open condition

I am reading the book "Arithmetic Dynamics" from Joseph Silverman that has the following comment about the Fatou set: where the Fatou of a map means the Fatou set of its iterations. I could ...
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### Prove $(f_n) = (n(\exp(x/n)−1))$ is pointwise convergent, pointwise bounded and equicontinuous on $[0,1]$

For n ∈ $\mathbb{N}$, let $f_n : [−1, 1] \rightarrow \mathbb{R}, f_n(x) = n(\exp(x/n)−1)$. a) Show that the sequence of functions $(f_n)$ converges pointwise on $[−1, 1]$, and determine the limit ...
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### Show that the sequence of functions converges uniformly.

I've had issues solving the following problem. The closest I have come is shown below. The problem is that in my "proof", I'm not using the fact that $K$ is compact, which makes me ...
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### Determine which sequences are equicontinuous.

I'd appreciate if somebody could check if my proofs below are correct and also give me some hints on the equicontinuity of $(k_n)$ (part c)). $\mathbb{R}$ represents the real line with the standard ...
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### An equicontinuous sequence of functions converging in $L^1$ converges pointwise a.e.

Let $f_n$ be a sequence of continuous functions defined on $\{x\in \mathbb{R}^d:|x|\leq 1\}$ such that $\|f_n\|_{1}\to 0$. Further, suppose that $f_n$ is equicontinuous. I need to show that $f_n\to0$ ...
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### Union of two equicontinuous families of functions is equicontinuous

Let $X$ be a metric space, $\mathcal{F}$ and $\mathcal{G}$ are families of real valued functions. Suppose that $\mathcal{F}$ and $\mathcal{G}$ are equicontinuous. I ever read that union of two ...
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### If $\{ f_n\}$ is equicontinuous on $X$ and bounded on dense set $D\subseteq X$, then $(f_n)$ is convergent on X.

Suppose $(X,d)$ complete metric spaces, $D\subseteq X$ dense in X, and $\mathcal{F}=\{f_n:X\to \mathbb{R}: n\in \mathbb{N}\}$ is equicontinuous on $X$. If $\mathcal{F}$ is bounded (above, below) on $D$...
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### Uniform convergence to zero implies equicontinuity

I am trying to understand the following explanation: 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 Particularly the ...
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### Does a sequence of $C^1$ functions converge to a $C^1$ function if the derivatives are bounded and equicontinuous?

My question is: If $f_n\to f$ pointwise in $(0,1)$ where $f_n\in C^1$ (continuously differentiable), AND the family $\{f_n'\}$ is uniformly bounded and equicontinuous, then is $f\in C^1$ as well? Ie, ...
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In a past exam of a course I am taking this semester there is the following question for which I can't find any example by myself. The question is to find an uniformly equicontinuous sequence $(f_n)$ ...
### Given a bounded sequence in $L^1([a,b])$, is $(t\mapsto \int_a^t f_n \,\mathrm d\lambda)_n$ equicontinuous?
We are given a bounded sequence $(f_n)_{n\in \mathbb N}$ in $L^1([a,b])$. This means there is some $M>0$ such that for all $n\in\mathbb N$, $\int_{[a,b]} |f_n|\,\mathrm d\lambda \leq M$. I wonder ...