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".
1
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0answers
12 views
An element of a set with a finite cover must be an element of at most two open intervals in a subcover?
Prove:
If a set $A\subseteq\mathbb{R}$ has a cover consisting of a finite number of open intervals, then A has a subcover such that for each $x\in A$, x is an element of at most two of the open ...
1
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2answers
27 views
Is $ f \colon (1, +\infty) \to \mathbb{R} , f(x) = \sin \frac{1}{x} $ uniformly continuous?
Is $$ f \colon (1, +\infty) \to \mathbb{R} ,
f(x) = \sin \frac{1}{x} $$ uniformly continuous? I think it could be but can't prove it. Would appreciate the help.
Edit: I'm looking for an answer in ...
1
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1answer
11 views
Problem: Is function unifomly continuous?
Is sin ( 1/x ) uniformly continuous on set (1, + infinity) ?
I tried to prove that it is Lipschitz continuous but I got stuck.
-3
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1answer
44 views
Is Openess and closedness are preserved under uniformly continuous map? [on hold]
Question: does uniformly continuous function maps open set to open set? and does uniformly continuous function maps closed set to closed set?
"I think they are false" but I couldn't able to find ...
0
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1answer
12 views
Is $ f \colon \mathbb{R^2} \to \mathbb{R^2} , f(x,y) = ( x^2, y^2) $ uniformly continuous on $\mathbb{R^2}$
Is $ f \colon \mathbb{R^2} \to \mathbb{R^2} , f(x,y) = ( x^2, y^2) $ uniformly continuous on $\mathbb{R^2} ?$
I think it is not, I tried proving it by contradiction but can't find the right $\delta$
0
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0answers
42 views
Prove that $f$ is well defined and is differentiable in $[0,\infty)$. [on hold]
The function is
$$f(x)=\displaystyle\sum_{n\geq1} \dfrac{x}{\sqrt n \ (x+n) }$$
2
votes
1answer
32 views
Is there a bounded rational function that is not uniformly continuous on $\mathbb{R}$?
I think the answer is no. I tried looking for one hour and it didn't work. Isn't every rational bounded function uniformly continuous ? There's nothing online about them.
1
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2answers
57 views
Is $f(x) = \frac{1}{x}$ uniformly continuous on $(1, \infty)$? [on hold]
is the function $f(x) = 1/x$ uniformly continuous on $(1, \infty)?$ I know that it is not uniformly continuous on $(0, \infty)$, but now I'm restricting it even more to get rid of most of the bad ...
0
votes
2answers
35 views
Using the sequential definition of uniform continuity to show $\sin(x)$ is uniformly continuous on $\mathbb{R}$
I want to show $\sin(x)$ is uniformly continuous on $\mathbb{R}$. Let $\{a_{n}\}$ and $\{b_{n}\}$ be sequences such that $\lim_{n\to\infty}[b_{n} - a_{n}] = 0$. Then, we need to show $\lim_{n\to\infty}...
1
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1answer
46 views
Is it true that $f(x) = x^{2}$ is uniformly continuous on $\mathbb{N}$?
I think that the answer is yes. If $f(x) = x^{2}$ were uniformly continuous on $\mathbb{N}$, then for every pair of sequences $\{u_{n}\}$ and $\{v_{n}\}$ satisfying
$$\lim_{n\to\infty} \left( u_{n} - ...
1
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0answers
50 views
Consider the series $\sum_{n=1}^\infty \frac{(-1)^n}{n+x}$
Consider the series
$$\sum_{n=1}^\infty \frac{(-1)^n}{n+x}$$
Find all $x \in \mathbb{R}$ at which the series converges. Converges absolutely. Find all intervals of $\mathbb{R}$ where the series ...
0
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1answer
26 views
Proof that if f(a)<0 and f(b)>0 f and is continuous on [a,b] then f changes sign at some c in (a,b) Part 2
I asked a question similar to this previously but I realized that what I was trying to prove was false. Then I changed the thing I was trying to prove but it was still false. I also accepted an answer ...
2
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1answer
19 views
$f(x)g(x)$ is uniformly continuous if $f,g$ are uniformly continuous and $\lim_{x\to \infty}g(x)=0$?
on $[0,\infty)$, can we prove $f(x)g(x)$ is uniformly continuous if $f,g$ are uniformly continuous and $\lim_{x\to \infty}g(x)=0$?
If $\lim_{x\to \infty}g(x)=0$ is droped, then it is easy to see that ...
1
vote
1answer
41 views
Let $ f : X \to Y $ a biyective funtion and uniformly continuous. [on hold]
Let $ f : X \to Y $ a biyective funtion and Uniformly continuous.Prove that is Y is complete and $f^{-1}$ is continuous, then X is complete.
0
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1answer
31 views
What is the best that you can say about a function with given property
Let $E \subset \mathbb R$, and $f:E\rightarrow \mathbb R$, be a map. Set $$\Delta_{\varepsilon,a} = \{\delta>0: B(f(a),\delta)\supset f(E\cap B(a,\varepsilon))\}\ for\ \varepsilon>0,a\in E$$
$$...
1
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0answers
16 views
Does unbounded and unbounded derivative imply not uniformly continuous?
Perhaps a more precise wording of the statement I'm trying to prove says something like this: Let $I$ be an interval in $\mathbb{R}$, and allow for the possibility that $I$ is unbounded. If a ...
1
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0answers
26 views
A uniformly continuous function on a bounded set whose image is not bounded? [duplicate]
I'm trying to find an example of a uniformly continuous function $f:X\to Y$ such that $f(X)$ is not bounded, for some bounded metric space $X$ and a metric space $Y$.
The classic exercise is to show ...
2
votes
2answers
125 views
Pointwise product of uniformly continuous functions
True or false: Let $f(x)$ and $g(x)$ be uniformly continuous functions from $\mathbb{R}$ to $\mathbb{R}$.
Then their pointwise product $f(x)g(x)$ is uniformly continuous.
I think it will be true; ...
1
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0answers
19 views
A property of modulus of continuity
I'm trying to show this fact, for a generic function $f:E\subset\mathbb{R}\to\mathbb{R}$:
$$ \omega(\delta _1+\delta_2)\leq \omega(\delta _1)+\omega(\delta _2) $$
where $$\omega(\delta):=\sup_{x_1,...
0
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1answer
42 views
What is the difference between “for every $x$ there exists a $y$” and “for all $x$ there exists a $y$”?
Is there any difference between them? or are they same?
I was going through the definition of uniform continuity, which uses the term "for every". Can someone explain me the difference between the two ...
0
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0answers
28 views
Give an example of a set $S$ whose support function is not continuous on $R^n$.
Let $A \subseteq \mathbb{R}^n$.
The support function of set $A$ is defined as the following
$$
S_A(x)=\sup_{y \in A} x^Ty
$$
where $x \in \mathbb{R}^n$.
Notice that when $A$ is bounded the function ...
1
vote
1answer
48 views
Show that a uniformly continuous function on $[0, 1)$ must be bounded.
Suppose that $f: [0, 1) \to \Bbb{R}$ is uniformly continuous. Show that there exists $N \in \Bbb{N}$ such that for every $i \in$ {$1, 2, ..., N$} and every $x, y \in [\frac{i-1}{N}, \frac{i}{N}), |f(x)...
1
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1answer
31 views
How to prove that this function is continuous but not uniformly continuous?
I am having some troubles solving a question on my homework sheet:
Prove that $\,\,f:\Bbb{R}^n\setminus\{0\}\to \Bbb{R}^n$, defined as $$f(x) = \frac{x}{||x||}$$ is continuous but not uniformly ...
-1
votes
2answers
29 views
Why does a uniformly continuous function on [a,b] in the reals need be closed AND bounded?
I cannot come up with any counter examples as to why a function would need to be both closed and bounded to be uniformly continuous. Why is it not sufficient to just have one condition?
For example, ...
2
votes
1answer
68 views
$\epsilon-\delta$ proof that $f(x) = x \sin(1/x)$ on $(0,1)$ is uniformly continuous
I have to prove
$$
f: (0,1) \to [-1,1], \ \ f(x)=x\sin(1/x)
$$
using $\epsilon-\delta$.
Try
Fix $\epsilon >0$. I have
$$
\left|x \sin(1/x) - y \sin (1/y)\right| \le |\sin (1/x)| |x-y| + |y| |\...
1
vote
2answers
43 views
Show that the support function of a bounded set is continuous.
The support function of a set $A \in \mathbb{R}^n$ is defined as the following
$$
S_A(x)=\sup_{y \in A} x^Ty
$$
where $x \in \mathbb{R}^n$.
Show that the support function of a bounded set is ...
2
votes
1answer
22 views
Uniform continuity on a union of sets
Let $f: X \rightarrow Y$ be a function on metric spaces. Let $E_{1}, ..., E_{n}$ be a finite collection of subsets of $X$, a metric space, such that $d(x,y) >1$ whenever $x \in E_{i}$ and $y \in E_{...
0
votes
1answer
45 views
If $f:\Bbb{R}\to\Bbb{R}$ is uniformly continuous, then $f$ is bounded
I have proved :
If $f$ is uniformly continuous on $(a,b)$, then $f$ is bounded on $(a,b)$.
But when it extends $(a,b)$ to $\Bbb{R}$, I don't know how to continue. Would you please give some ...
9
votes
1answer
127 views
What kind of “geometric” regularity $f'^2$ gives on $f$
When solving real-analysis' problems I like to represent the functions involved and think geometrically what is going on.
Today I got the following exercise :
Let $f \in \mathcal{C}^1(\mathbb{R},\...
0
votes
0answers
24 views
Proving Uniform Continuity of Fourier transform (Related to measure theory)
Let $\mu$ be a finite measure on $(\mathbf{R},\mathcal{B})$. Define $\hat\mu(t)=\int{e^{-itx} d\mu(x)} , t\in \mathbf{R}$. Show that the function $\hat\mu$ is uniformly continuous on $\mathbf{R}$. $\...
0
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1answer
23 views
Reverse Lipschitz condition
Assume that $f:[a,b]\to\mathbb{R}_+$ is continuous, differentiable with f'(x)>0 and Lipschitz.
For $y>x$, I want to understand if
$$
f(y)-f(x) \ge L (y-x)
$$
holds true where $L$ is a strictly ...
-2
votes
0answers
21 views
Uniform Continuity Question…
There is a question on an assignment that I am currently doing and it states
(a) Let $0<\theta<\frac{\pi}{2}$ and $f(x)\neq 0$ on $[0,\theta]$ where $f$ is continuous on $[0,\theta]$. Show that ...
2
votes
1answer
36 views
Bounded Derivative Test
I'm currently working on a question in an assignment, and it asks us to use the bounded derivative test, to determine whether $$f(x)=\sin\left(\frac{1}{x}\right)$$ is uniformly continuous on the ...
2
votes
0answers
31 views
Partial derivative under integral sign
My professor gave me to prove this statement:
Let $A(t,x) \in C^1([0,1]\times\Bbb R, \Bbb R).$
Then $\displaystyle \frac{\partial}{\partial x}\int_0^1A(t,x)\ dt = \int_0^1\frac{\partial A}{\...
0
votes
1answer
18 views
Why does this argument that sin(1/x) is not uniform continuous on (0,1) fail for [0,1]?
(1): I understand that $f(x) = sin(1/x)$ is continuous at every point in the open interval (0,1), but it is not uniformly continuous on this interval (e.g. take $\epsilon=2$ and set $x_n = \frac{1}{\...
0
votes
1answer
22 views
Proof Question: Continuous Functions on Compact Domains are uniformly continuous
So, I'm going over the proof of the following theorem in Real Analysis, but I am unable to comprehend a step in the proof. The theorem is as follows:
$\textbf{Theorem:}$ Let $f$ be a continuous ...
1
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2answers
58 views
Let $f:(a,b) \to \mathbb{R}$ to be differentiable and unbounded. Prove: $f'$ isn't bounded on $(a,b)$
Let $f:(a,b) \to \mathbb{R}$ to be unbounded and a differentiable function.
Prove: $f'$ isn't bounded on $(a,b)$
My Attempt:
suppose towards contradiction that f' is bounded in $(a,b)$.
then ...
0
votes
2answers
38 views
How can a continuous function $f:[0,1] ^{\mathbb{Q}}\rightarrow \mathbb{R}$ not be uniformly continuous?
If the domain $[0,1] ^{\mathbb{Q}}$ of a continuous function $f$ consists of all rationals between zero and one, inclusive, how can the function not be uniformly continuous?
From my understanding, a ...
1
vote
1answer
26 views
a problem about continuous functions on closed intervals
This is the problem:
Let f be a continuous function on a finite interval [a,b]. Suppose that f (x) > $0$ for all x in [a,b]. Prove that there is an $\alpha$ > $0$ such that f (x) > $\alpha$ for ...
2
votes
0answers
23 views
Difference between uniformly continuous, absolutely continuous and Lipschitz continuous functions? [closed]
Can somebody please explain me the difference between uniform continuity, absolutely continuity and Lipschitz continuity using examples and counterexamples?
0
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1answer
34 views
$f$ is uniformly continuous on $[a,b]$ and that $g$ is uniformly continuous on $g([a,b])$. Then $f\circ g$ is uniformly continuous on $[a,b]$
Suppose that $f$ is uniformly continuous on $[a,b]$ and that $g$ is uniformly continuous on $g([a,b])$ . Then $f\circ g$ is uniformly continuous on $[a,b]$.
Could anyone give me a hint for answering ...
1
vote
1answer
46 views
Confusion in a Proof
The question and its proof are given below:
$ \text{Page 86, Problem 12} $
$ \color{red}{\text{A function $ f $ is said to be periodic with period $ p $ if $ f(x + p) = f(x) $ for all real $ x ...
0
votes
2answers
24 views
Determine whether the function $f(x) = \sqrt {x}$ is uniformly continuous on $A = [1, +\infty)$.
Determine whether the function $f(x) = \sqrt {x}$ is uniformly continuous on $A = [1, +\infty)$.
The answer of the question is given below:
But I could not understand why $\sqrt{x} + \sqrt{y} &...
1
vote
0answers
25 views
Is there any flaw in a given proof of letter (a)?
The question and its answer is given below:
I prefer the above proof than the below one:
Is there any flaw in the first proof?
0
votes
1answer
25 views
Is the function $f(x) = e^x \cos (1/x)$ is uniformly continuous on $A = (0,1)$. [duplicate]
The question and its answer is given below:
Determine whether t the function $f(x) = e^x \cos (1/x)$ is uniformly continuous on $A = (0,1)$.
but I am wondering why the $n$ is chosen like this, ...
1
vote
1answer
40 views
Determine whether the function $f(x) = e^x \cos (1/x)$ is uniformly continuous on $A = (0,1)$.
Determine whether the function $f(x) = e^x \cos (1/x)$ is uniformly continuous on $A = (0,1)$.
I see that the function $f(x) = e^x \cos (1/x)$ is uniformly continuous on $A = (0,1)$. Am I correct?
...
1
vote
1answer
27 views
If f is a uniform continuous function on (a,b), then f is bounded on (a,b).
If $f$ is uniformly continuous on $(a, b)$, then $f$ is bounded on $(a, b)$.
I'm trying to understand this proof given here in this link. Its the second proof given from the question in the link, ...
2
votes
0answers
31 views
Extraneous Condition in the Hypothesis?
If $\{f_n\}$ is a sequence of continuously differentiable functions on $[a,b]$, $f_n \to f$ uniformly on $[a,b[$, and there is a function $g : [a,b] \to \Bbb{R}$ such that $f'_n \to g$ uniformly on $[...
0
votes
2answers
35 views
Non uniform continuity of $f(x)=x^3$ on the interval $[10,\infty)$
How to show that $f(x)=x^3$ is not uniform continuous on the interval $[10,\infty)$?
I am well aware that $f(x)^3$ is not uniform continuous on the set of real numbers but and I believe on $[10,\...
0
votes
0answers
19 views
How does uniform continuity and continuity affect the convergence of a function
Say we are working in $L^{\infty}(\mathbb{R})$
In the below cases what can we say about the convergence?
1) $f_{n},f$ are continuous and $\lim_{n\to \infty}f_{n}=f$
2) $f_{n},f$ are uniformly ...
