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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".

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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 $f(x)=x$.
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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 ...
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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\...
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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 ...
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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[...
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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 ...
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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 ...
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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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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 ...
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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-...
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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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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 ...
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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 $|...
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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 ...
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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 $cost_k^2$=-...
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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 ...
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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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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 $$ f \colon X \to f(X), f(x) =x^3 $$ ...
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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 ...
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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}...
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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/...
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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 ...
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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 ...
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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)^{...
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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 ...
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 $A,B$ ...
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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 ...
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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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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 ...
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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 ...
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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 &...
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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 $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
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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
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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)...
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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{|...