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".
0
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1answer
66 views
Is $f(x)=x$ uniformly continuous on $\mathbb{R}$? [on hold]
Is $f(x)=x$ uniformly continuous on $\mathbb{R}$?
I know that $x^2$ is not uniformly continuous on $\mathbb{R}$ but not too sure about $f(x)=x$.
1
vote
1answer
49 views
Continuity on infinity
We have $f: [a.b] \rightarrow \mathbb{R}$ continuous, and
$$c_n = \sup\{c \in [a,b] : |f(x) - f(c_{n-1})| < \epsilon \text{ for any } x \in [c_{n-1},c]\}$$
with $c_1 = a$. In a previous exercise ...
1
vote
1answer
33 views
Rephrasing the “if” part of the statement of the Principle of Uniform Boundedness
I am trying to better and more rigorously understand the Principle of Uniform Boundedness (PUB). Recall that the statement of PUB reads:
Let $X,Y$ be Banach Spaces and suppose that $(T_n)_{n\in\...
0
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1answer
45 views
Whether the product of uniformly continuous functions is uniformly continuous [closed]
I know it isn't and I have to give a counter-example.
Function $f_1(x)=f_2(x)=x$
this is a uniformly continuous function
the product of these functions $f_1(x)\cdot f_2(x)=x\cdot x=x^2$ this isn't an ...
4
votes
3answers
83 views
Log (uniform) continuous functions
I am interested in functions $f: \mathbb R_+\to \mathbb R_+$ such that $\log \circ f \circ \exp$ is uniformly continuous. In other words
\begin{align}
\forall_{c}\, \exists_{c'}, \forall_{ x,x'\in[...
0
votes
2answers
18 views
Showing |f(x)-f(y)| < $(\frac1\epsilon)^2*|x-y|$ for x,y on [$\epsilon^3/4,1$]
f(x) in this case equals $x^\frac13$
So far I've tried setting $|x-y| < \delta$, with $\delta$ = 2$\epsilon^3$, therefore making $(1/\epsilon^2)|x-y| < 2\epsilon$, but this doesn't show that ...
0
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1answer
24 views
Contraction and $\max$ function
$f: \Bbb R \mapsto \Bbb R$
$g: \Bbb R \mapsto \Bbb R$
$h: \Bbb R \mapsto \Bbb R$
$h:=\max\{f(x), g(x)\}$
Is $h$ a contraction on $ \Bbb R$ if $f$ and $g$ are both so?
First attempts of ...
4
votes
2answers
102 views
Is $f(x)=\frac{1}{x+1} \cos x^2$ uniformly continuous?
Let $f:[0,\infty)\to \Bbb{R}$, $f(x)=\frac{1}{x+1} \cos x^2$, Is $f$ uniformly continuous?
My attempt:
Let $x,y\in [0,\infty),$ then
\begin{align*}
\Bigg|\frac{\cos x^2}{x+1}-\frac{\cos y^2}{y+1}\...
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votes
0answers
19 views
Cantor function counter example to Heine theorem?
I have a problem with Heine-Theorem, which says that any continous function from a compact metric space to a metric space is uniformly continous.
How does it not apply to Cantor function which is ...
0
votes
1answer
28 views
Probability of nth event is between x and y
I have a uniform distribution of age in the range $[a, b)$ with $a=42$ and $b=78$
So the probability that a person walks in a bank that is between $50$ and $70$ years of age would be $\frac{70-50}{78-...
0
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0answers
53 views
Example of a continuous function
Give an example of two open sets A, B and a continuous function $f:A\cup B\rightarrow\mathbb{R}$ such that $f|A$ and $f|B$ are uniformly continuous but $f$ is not.
I have been stuck in this one for a ...
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votes
1answer
32 views
If $f$ is continuous on $\mathbb R$ and limits to $\infty$ ,$-\infty$ exist, prove that $f$ is uniformly continuous
Let $f:\mathbb R\to \mathbb R$ be a continuous function such that $\lim_{x\to \infty}
\> f(x)$ and $\lim_{x\to -\infty} f(x)$ exist and are finite. Prove that
$f:\mathbb R\to \mathbb R$ is ...
0
votes
1answer
23 views
Is $h(x)=x^\frac{3}{2}$, $D_h=[0,\infty)$ uniformly continuous?
Is $h(x)=x^\frac{3}{2}$, $D_h=[0,\infty)$ uniformly continuous ?
My attempt:
i) since $h$ is continuous on $[0,2]$ then it is uniformly continuous on $[0,2]$.
ii) If $x,u \in [1,\infty)$, then
$|...
-2
votes
0answers
37 views
What is intuition behind uniform continuity?
What is intuition behind uniform continuity ?
Actually I want to know the geometrical approach .If we discuss uniform continuity on $(- \infty ,\infty ) ,(-\infty,-1),(0,\infty)$ and $[0,\infty] $ of $...
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4answers
57 views
Is $\sum\limits_{n=1}^\infty \arctan \frac x {n^2}$ continuous?
Let
$$
f(x) = \sum\limits_{n=1}^\infty \arctan \frac x {n^2}
$$
I need to check whether $f : \mathbb R\to \mathbb R$ is continuous.
Of course, if it converges, $f(x) = -f(-x)$, so I will be only ...
0
votes
1answer
36 views
f(x)=cos(x^2) is not uniformly continuous
Prove that $f(x)=cos(x^2)$ is not uniformly continuous on the set R.
I started out by solving $cos(x_k^2)$=1⟹cosX=1 ⟹ $X=2_k\pi$
$x_k^2$=$2k\pi$=>$x_k$⟹$\sqrt(2k\pi)$ -> goes to infinity
$cost_k^2$=-...
1
vote
1answer
19 views
Metric spaces problem regarding if composition function is uniformly continuous then what about individual function itself
Suppose $X, Y, Z$ are metric spaces and $Y$ is compact. Let $f$ maps
$X$ into $Y$. Let $g$ be continuous one-to-one map $Y$ into $Z$ and
put $h(x)=g(f(x))$ for $x$ in $X$. If $h(x)$ is uniformly ...
0
votes
1answer
52 views
Prove that f is uniformly continuous
I found this example of uniformly continuous but had a question regarding a step.
Let $S=R$ and $f(x)=3x+7$. Then $f$ is uniformly continuous on $S$.
Proof: Choose $\epsilon > 0$. Let $\delta= \...
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votes
2answers
51 views
Is the function $ f \colon X \to f(X), f(x) =x^3 $ continuous, uniformly continuous
Let $$ X = \big( [0,1] \cap
\mathbb{Q} \big) \cup \left\{ 1+ \frac1n \middle|\ n \in \mathbb{N} \right\} $$ be a subspace of $ \mathbb{R} $. Is the function $$ f \colon X \to f(X), f(x) =x^3 $$ ...
1
vote
0answers
25 views
How to demonstrate the uniform continuity of a function in the case it has no limited derivative
First of all, I would like to apologize if I make grammatical errors because English is not my mother tongue.
During a lecture at the university my teacher did an exercise that required to find the ...
0
votes
1answer
43 views
Proving $1/ x^2$ is not uniformly continuous on (0,2]
I am trying to prove the function in the title is not uniformly continuous on (0,2].
Different from the proof given in proof, I proceed to show the proof following lecture as follows:
\begin{equation}...
0
votes
1answer
22 views
$((x/c)^n)$ has no convergent subsequence in C[a,b]
Consider the metric space C[a,b] equipped with the metric $d(f,g)=max|f(x)-g(x)|$. I would like to show that the sequence $((x/c)^n)$ has no convergent subsequence, where c is a constant such that $x/...
3
votes
3answers
175 views
Uniform continuity implies existence of limit of integral
Let $f: (0,1) \rightarrow \mathbb{R} $ be uniformly continuous. Prove that $$ \lim_{\epsilon \to 0} \int^{1-\epsilon}_{\epsilon}\!\!f(t)dt \in \mathbb{R}$$ Any ideas?? $f$ can be extended to a ...
1
vote
0answers
14 views
Any continuous mapping on compact set is uniformly continous
I've got some problems while I'm proving that statement.
I'm trying to show that with using RAA(it is PMA exercise Problem 10, chapter 4).
Here is my idea.
Suppose $f$ is continuous but not ...
0
votes
1answer
26 views
Which of the conditions imply that a function is uniformly continuous relative to an uniformity?
Let $(X, \mathcal{U})$ be an uniform space. $f:(X, \mathcal{U})\to (X, \mathcal{U})$ is called uniformly continuous relative to $ \mathcal{U}$, if for every entourage $V\in \mathcal{U}$, $(f\times f)^{...
1
vote
1answer
41 views
$f$ is continuous function on $[0, \infty)$, and the limit $\lim_{n \to \infty} \frac{f(x)}{x}= a \in \mathbb{R}$ exists. $f$ is uniformly continuous?
I'm not sure how to prove or give a counter example to this. I wasn't able to prove it but I couldn't think of any counter example. Here's what I tried: By definition there exists $M>0$ such that ...
1
vote
1answer
60 views
Proving uniform continuity of $f(x) = \sqrt{1-x^2}$ on $[-1,1]$
I want to prove that $f(x) = \sqrt{1-x^2}$ is uniform continuous on the interval $[-1,1]$. Let $f(x) = \sqrt{1 - x^2}$. Then I need to show:
$\forall \epsilon > 0 \enspace \exists \delta > 0 \...
-2
votes
1answer
106 views
If $f(x+1)=f(x)$ then?
Let $f: \ \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+1) = f(x)$, $\forall x \in \mathbb{R}$.
Then which of the following statement(s) is/are true?
$f$ is bounded.
$f$ is bounded ...
0
votes
3answers
115 views
How to prove that $\sqrt x$ is continuous in $[0,\infty)$?
I am trying to prove that
$\sqrt x$ is continuous in $[0,\infty)$.
I have started writing the following proof:
Given $x_0 \in [0,\infty)$ and $\epsilon > 0$. We have to show that there exists ...
1
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1answer
47 views
Is every uniformly continuous function 1-1 and onto?
Let $f : (X,d)\rightarrow (Y,\rho)$. Is $f$ 1-1 and onto if $f$ is a uniformly continuous function on X?
If not, would $X$ being compact change things?
If not, do you know a theorem or something ...
0
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0answers
39 views
Existence of a uniform continuous function $f$ s.t. $\displaystyle \sum_{n=1}^{\infty}\frac{1}{f(n)}$ converges. [duplicate]
TRUE/FALSE
There exists a uniformly continuous function $f:(0,\infty)\to (0,\infty)$ such that $\displaystyle \sum_{n=1}^{\infty}\frac{1}{f(n)}$ converges.
I have no idea how to find an example ...
0
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2answers
64 views
Prove that $f(x)=x\sin(1/x)$ for $x\ne0$, $f(0)=0$, is not Lipschitz on $[0,1]$
Prove that $f(x)=\cases{0& if $x=0$\\x\sin(1/x)& otherwise,}$ is not Lipschitz on $[0,1]$
MY TRIAL
My idea is to show that $f$ does not have a bounded derivative.
So, suppose for ...
4
votes
1answer
48 views
Uniform continuity and compactness
We know that, if a function $f$ is continuous mapping from a compact metric space to another metric space say $Y$ then $f$ is uniformly continuous.
Do we have a generalization of this theorem for ...
0
votes
2answers
53 views
Probability Distribution Table of Bus Arrival
For an assignment in my high school Data Management class, I am required to make a probability distribution table for my data, which is "Wait Time for a Bus." It is not just any bus. It is a specific ...
0
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1answer
35 views
Confusion regarding Cauchy's General Principle and Uniform Convergence
The definitions of the two are so alike, that it confuses me.
Cauchy's General Principle:
The necessary and sufficient condition that a function $f(x)$ may tend to a definite limit, say $l$, as $x ...
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0answers
17 views
limit at infinity quotient by x privided given function uniformly continuous
Suppose $f(x)$ is real valued uniformly continuous function on $ \mathbb R$. Is it true that limit of $f(x)/x$ exists as x tends to infinity and minus infinity?
2
votes
1answer
45 views
Possible characterization of compact metric spaces via real-valued uniformly continuous functions?
1) If $X$ is a metric space such that the image of every uniformly continuous function $f: X \to \mathbb R$ is bounded, then is it necessarily true that $X$ is compact ?
2) If $X$ is a metric space ...
6
votes
2answers
58 views
On a special type of sequence in complete metric space
Let $\{x_n\}$ be a sequence in a complete metric space $X$ such that for every uniformly continuous function $f:X \to \mathbb R$, the sequence $\{f(x_n)\}$ is convergent in $\mathbb R$. Then is it ...
3
votes
1answer
39 views
On a version of Urysohn lemma for complete metric spaces, involving uniform continuous functions
Let $X$ be a complete metric space. Let us say that given a subset $A\subseteq X$, a point $a\in X$ is a limit point of $A$ if for every $r>0, A \cap B(a,r)\setminus \{a\} \ne \phi$. Now let $A,B$ ...
1
vote
3answers
63 views
$\frac{1}{x}$ not uniformly continuous
In my textbook we saw an example of a not uniformly continuous function, $f(x) = \frac{1}{x}$ but i find the explanation why kinda weird. First of all, this is the definition of uniform continuity in ...
2
votes
2answers
54 views
Showing that $f$ is a uniformly continuous function on the interval $[0, \infty)$
Suppose that $f$: $[0, \infty) \rightarrow \mathbb{R}$ is continuous on $[0, \infty)$ and differentiable on $(1, \infty)$ with bounded derivative. Show that $f$ is uniformly continuous. (HINT: split $...
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vote
1answer
70 views
If $f$ is $C^{1}$ and $\sup_{x \in \mathbb{R}}|f'(x)| = \infty$ , prove that $f$ cannot be uniformly continuous
I'm looking for help with a proof that I still cannot figure it out. Here is the statement:
"If $f:\mathbb{R} \rightarrow \mathbb{R}$ is $C^{1}$ and $\sup_{x \in \mathbb{R}}|f'(x)| = \infty$ , prove ...
-1
votes
1answer
59 views
Proving uniform continuity of $f(x) = x \sin{\frac{1}{x}}$ on $(1,2)$ directly from the definition.
Before this is marked as a duplicate, I have already looked at the answers here and here.
I believe my question is different because I want to do this proof directly from the definition. The proof ...
0
votes
0answers
36 views
Uniform continuity of $f(x)=\sum_{n=1}^\infty f_n(x)$ on $\mathbb R$
Suppose for $n\in \mathbb N$, $f_n(x) =
\begin{cases}
n(x-n+\frac{1}{n}) & \text{if $x\in [n-\frac{1}{n},n]$} \\
n(n+\frac{1}{n}-x) & \text{if $x\in [n,n+\frac{1}{n}]$} & \\ 0 &...
2
votes
2answers
99 views
Do the metrics $d$ and $\frac{d}{1+d}$ induce the same uniformity?
Let $d_1$ and $d_2$ be two metrics on the same set $M$. Then $d_1$ and $d_2$ are called uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
0
votes
1answer
35 views
On uniform convergence of function
This is a problem I have encountered on my exam. It is probably an easy one but after an hour of trying I could not come with an answer. If you could give me a small hint (just a starting point) that ...
1
vote
1answer
143 views
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous then it is uniformly continuous
I do have a general proof for this problem. That is if there is a continuous function on a compact interval then the function is uniformly continuous.
But I'm wondering whether there is a different (...
1
vote
2answers
48 views
If a function is uniformly continuous restricted to each line, is it globally uniformly continuous?
Question: Suppose $f: \mathbb{R}^2\to \mathbb{R}$ is a continuous function such that for every line $L$ passing through the origin $(0, 0)$, the restriction of the function $f|_{L} : L\to\mathbb{R}$ ...
0
votes
1answer
41 views
Uniform Continuity of sum of a series of functions
Let $f(x)=\sum_{n=1}^{\infty}f_n(x)$, where $$f_n(x)=\begin{cases}n(x-n+\frac1{n}),\ \ x\in[n-\frac1{n},n]\\n(n+\frac1{n}-x),\ \ x\in[n,n+\frac1{n}]\\0,\ \ \text{otherwise}\end{cases}$$
Then, is $f(x)...
0
votes
1answer
32 views
Determine if the function is uniformly continuous
Determine if the function: $$f(x)=\frac{x}{(1-x)^2}$$ $\forall x>1$ is uniformly continuous.
I know that it is not. But I'm having trouble proving it.
I reach to a point that i get this: $$\frac{|...
