# 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 $d_X (x_0, x) <\delta$.

151 questions
Filter by
Sorted by
Tagged with
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 ...
39 views

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 ...
23 views

### 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 ...
18 views

### $f_n$ 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. ...
41 views

140 views

42 views

### 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 ...
24 views

### 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 ...
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 ...
28 views

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 ...
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. ...
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 ...
22 views

### 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
60 views

### 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 ...
32 views

### 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 ...
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 ...
55 views

### 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 ...
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 ...
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 $x$ ...
23 views

26 views

### 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 ...
19 views

### 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 ...
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 ...
40 views

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 $T: (C([0,1]),\vert\vert \cdot \vert \... 0answers 242 views ### Tyrtyshnikov's proof that polynomial roots depend continuously on the coefficients I'm having trouble understanding the following proof, which is taken from A Brief Introduction to Numerical Analysis by Eugene E. Tyrtyshnikov. (Note: The polynomials in the following are complex) ... 3answers 313 views ### Is there a version of the Arzelà–Ascoli theorem capturing$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))$, I'... 1answer 42 views ### is the following sequence equicontinuous Let$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$be a subset of$\...
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 ...
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 ...