Questions tagged [uniform-continuity]
For questions involving the concept of uniform continuity, that is, "the $\delta$ in the definition is independent of the considered point".
2,129
questions
1
vote
0
answers
24
views
Finding a function in the unit sphere of a functional subspace with a couple of properties
Preliminaries:
A={f $\in C(X); f(a)=0$} is a banach space with norm the following:
$\Vert f\Vert=sup\vert f(x)-f(y)\vert; x,y \in X$
( X is Hausdorff and compact space. 'a' is a point in X)
$\tilde f ...
- 11
4
votes
2
answers
131
views
$\displaystyle \lim_{n \rightarrow +\infty} \dfrac{1}{\log(n)}\sum_{k=1}^n \dfrac{1}{k}f\left(\dfrac{k}{n} \right) = f(0)$
I am working on the following question : let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. Prove that : $$\lim_{n \rightarrow +\infty} \dfrac{1}{\log(n)}\sum_{k=1}^n \dfrac{1}{k}f\left(\...
- 63
0
votes
1
answer
28
views
Redundancy in the definition of uniform spaces
Let me paraphrase Wikipedia's definition of uniform spaces:
Definition A. A set $X$ endowed with a nonempty collection $\Phi$ of subsets of $X \times X$ is a uniform space if the following conditions ...
- 2,049
0
votes
0
answers
21
views
Uniformly Continuous and locally Lipschitz but not Globally Lipschitz Function on a "Connected but not compact" set
I know such a function exists but I can’t find an example. I have the famous example f:]0,inf[ --->R f(x)=sqrt(x) function. But I can't find any other function which is Uniformly Continuous and ...
- 51
0
votes
1
answer
49
views
Continuity question on compact and connected domain
Let $\Omega$ be an open and bounded connected domain from $\mathbb{R}^N$. Consider a continuous function $f:\overline{\Omega}\to\mathbb{R}$. So $K=\overline{\Omega}$ is compact.
My question is: Can we ...
- 1,912
0
votes
1
answer
55
views
Counterexample of function that is hard to find
Is there a continuous function $f:\left [0,\dfrac{1}{2}\right ]\to \mathbb{R}$ for which there is no constant $c>0$ such that:
$$|f(x)-f(y)|\leq\dfrac{c}{-\ln(|x-y|)},\ \forall\ x,y\in\left [0,\...
- 1,912
0
votes
2
answers
78
views
If $f'$ is periodic , are $f , f'$ uniformly continuous?
$f: ℝ \rightarrow ℝ $ is such that $f'$ exist and $f'$ is periodic. Are $f ,f'$ uniformly continuous ?
Attempt :- I was thinking along a counter example . What if $f'(x)=\tan x$ but then $f(x)=\ln (|...
- 2,589
1
vote
0
answers
36
views
Holomorphic and bounded implies uniform continuity
I'm working on the following Qualifying Exam problem:
Let $\mathbb{H}$ be the upper half plane, and let $f: \mathbb{H} \to \mathbb{C}$ be holomorphic and bounded. For a given $r> 0$, let $\mathbb{...
- 73
0
votes
1
answer
38
views
Relation between uniformly absolutely continuous and absolutely continuous in measure theory
I'm a little confused with these two definitions:
Definition of $\textbf{absolutely continuous}$:
$\nu$ is said absolutely continuous in respect to $\mu$ if $\mu(E)=0 \Rightarrow \nu(E)=0, \forall E \...
- 17
2
votes
4
answers
83
views
If $\lim\limits_{x\to +\infty} f(x)=L, (L\in \mathbb{R})$, $f'(x)$ is uniformly continuous on $(0,+\infty),$ then $\lim\limits_{x\to +\infty} f'(x)=0$
If $\lim\limits_{x\to +\infty} f(x)=L, (L\in \mathbb{R})$, $f'(x)$ is uniformly continuous on $(0,+\infty),$ then $\lim\limits_{x\to +\infty} f'(x)=0$.
$\textbf{Attempt}$:
$\textbf{Claim}$: If $f'(x)$...
- 1,541
0
votes
0
answers
33
views
$\mathbb R^n$ version of proof that uniform continuity implies boundedness [duplicate]
I have seen a lot of questions on the site about the proof that if $f$ is uniformly continuous on an interval, then its range is bounded. However, I have not been able to find a question that ...
- 711
1
vote
1
answer
40
views
Continuity and uniform continuity on products
Let $X,Y,Z$ be metric spaces, and $F:X \times Y \rightarrow Z$ be a continuous function. Further, suppose that for each $x \in X, F_x: Y \rightarrow Z$ is uniformly continuous. Then, is the canonical ...
- 444
1
vote
1
answer
77
views
Rudin's Construction of Lebesgue Measure 2
I am currently studying Rudin's RCA book and I have a question about Theorem 2.20, where the author constructs Lebesgue measure on $\mathbb{R}^k$.
Here are the definitions and the notations I am ...
- 13
2
votes
1
answer
37
views
Prove that if $f$ is continuous, and has two asymptotes, then it is uniformly continuous (Argument check)
The exercise is the following:
Let $f:\mathbb{R} \rightarrow \mathbb{R} $ continuous such that
$\lim_{x\rightarrow+\infty} f(x) = \ell_1$ and $\lim_{x\rightarrow-\infty} f(x) = \ell_2$ for some ...
-2
votes
1
answer
71
views
How can I prove that if $f:[0,+\infty)\to[0,1]$ is continuous and increasing, then it is uniformly continuous? [closed]
The problem is the following:
Let $f:[0,+\infty)\rightarrow\mathbb{R}$ such that it is continuous, strictly growing and its image is contained in $[0,1]$, is uniformly continuous.
I've tried in ...