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 $d_X (x_0, x) <\delta$.
Questions tagged [equicontinuity]
251 questions
21
votes
3 answers
Need some help on a non-example of equicontinuity
In an attempt to better understand the definition of an equicontinuous family of continuous functions, I want to find a simple non-example.
My intuition says that the family $\{f_n\colon[0,1]\to\mathbb R\}_{n\in\mathbb N}$ given by $f_n(x)=x^n$ is…
James
- 211
11
votes
2 answers
When is the compact-open topology locally compact?
Let $X$ and $Y$ be topological spaces, and consider the compact-open topology on $C(X,Y)$, which is generated by open sets of the form
$$\{\text{continuous }f\colon X\to Y:f(K)\subseteq U\}\text{ for compact }K\subseteq X\text{ and open }U\subseteq…
Thomas Browning
- 4,541
10
votes
2 answers
a set of functions that are pointwise equicontinuous but not uniformly equicontinuous, supposing the domain of f is noncompact
Can anyone think of an example of such set of functions?(If domain is compact then pointwsie equicontinuity implies uniformly equicontinuous)
user3404321
- 185
10
votes
3 answers
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've seen that the following version is used…
0xbadf00d
- 14,208
9
votes
0 answers
Is $\sin(nx)$ equicontinuous on $[0,1]$?
Consider $f_n=\sin(nx),\,x\in[0,1]$. In order to show that this is not an equicontinuous family, take $x=0,y=\frac{1}{N}$ where $N \in \mathbb{N}$ can be arbitrarily large, so $\forall \, \delta>0 \rightarrow y<\delta$.
Now, if we consider…
Ilia
- 499
7
votes
1 answer
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 $\left\{ f_n \right\}$ is pointwise bounded on $E$ if…
Saaqib Mahmood
- 27,542
7
votes
3 answers
Uniformly convergent implies equicontinuous
I'm trying to prove that if I have a sequence of continuously differentiable functions $f_n$ that converge uniformly on $[a,b]$, then $\{f_n\}$ is equicontinuous for all $x_0 \in [a, b]$.
My idea is to use uniform convergence to deal with the "tail"…
Rebekah
- 1,037
6
votes
2 answers
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 $B\subset C(K)$ be compact. Prove that
$B$ is equicontinuousas…
Lee
- 451
6
votes
1 answer
Proof that pointwise equicontinuity on a compact subset of $\mathbb{R}$ implies uniform equicontinuity.
I have an idea of the proof of the above statement, but I'm not entirely sure if it's right. Any comments would be appreciated. This question was supposedly answered here but the answer doesn't address the question at all.
Work so far:
Suppose not,…
5
votes
1 answer
Uniform Integrability and relative compactness
I am trying to proof relative compactness in L2(0,1) for a specific set of functions
$(\phi_n)_{n \in \mathbb{N}}$ with following properties:
$\int_0^1 \phi(x) dx = 0 $
$||\phi_n^2||_{L1(0,1)} = 1 $
$(\phi_n)_{n \in \mathbb{N}}$ is uniformly…
Alphacache
- 53
5
votes
1 answer
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 topology.
Consider the following sequences of…
JustANoob
- 1,729
- 11
- 25
5
votes
1 answer
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 uniformly equicontinuous on $K$.
Proof from…
Peter_Pan
- 1,938
5
votes
0 answers
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)
Theorem 3.9.1 Consider a parametrized batch of…
Thorgott
- 17,265
5
votes
1 answer
Supremum conditions for equicontinuity
Suppose $\mathcal{E} \subset C_0([a, b], \mathbb R)$ is a family of functions.
Show that $g(x) = \{\sup f(x) : f \in \mathcal{E}\}$ is continuous does not imply that $\mathcal{E}$ is equicontinuous.
If $g(x) = \{\sup f(x) : f \in \mathcal{F}\}$ is…
Aden Dong
- 1,477
5
votes
1 answer
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 example. I saw this answer:…
Lucas
- 4,260
- 1
- 21
- 57