# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...
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 ...