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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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
34 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
57 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
98 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
84 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
46 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
51 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
43 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
47 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
25 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
13 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
43 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
56 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
33 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
49 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
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2answers
51 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
66 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
58 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
35 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
89 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
34 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
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1answer
142 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 (...
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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
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1answer
36 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{|...
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2answers
26 views
Identical Cauchy sequences and continuity.
Find a set $X$ and two metrics $d$ and $m$ on $X$ such that the Cauchy sequences of $(X,d)$ and $(X,m)$ are identical and the identity map from $(X,d)$ to $(X,m)$ is continuous but not uniformly ...
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1answer
46 views
Nearest point property and uniformly continuous image
Show that a uniformly continuous image of a metric space that has the nearest point property need not have that property.
I have some trouble understanding the problem.
With the term ''uniformly ...
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2answers
67 views
Prove tht $f(x) = x^{2}$ is uniformly continuous on $\bigcup_{n = 1}^{\infty} [n, n + 1/n^{3}]$.
How can I show that the function $f(x) = x^{2}$ is uniformly continuous on $\bigcup_{n = 1}^{\infty} [n, n + \frac{1}{n^{3}}]$?
As $n \to \infty$, I know that we get only all of the integers; ...
3
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1answer
44 views
How to prove $x^{1/n} $ is uniformly continuos in $[0, a]$ where $a$ is a positive real.
I proved that this funcion is not Lispchitz continuos making $y = 2x$ and making $x \rightarrow 0$. But I'm stuck proving the uniformly continuity.
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2answers
32 views
Is $f:(-1,1)\to R$ defined as $f(x)= \frac{x}{1-\lvert x \rvert}$ and $f^{-1}(x)$ uniformly continuos?
I think this fuction is uniformly continuos, but the inverse is not since the limit in the proximities to $\lvert x \rvert = 1$ the function has a vertical asymptote.
However I tried a lot of algebra ...
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1answer
29 views
Proving that a differentiable function f, with f' bounded, is uniformly continuous
Let $f: R \rightarrow R$ be a differentiable function. Prove that if $f'$ is bounded, then $f$ is uniformly continuous
My attempt:
Since $f$ is differentiable, we have $f'(x) = \lim_{x \to x_0}\...
4
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2answers
98 views
uniformly continuous function $f$ such that $\sum 1/f(n)$ is convergent?
Does there exist a uniformly continuous function $f:[1,\infty)\to \mathbb R$ such that $\sum_{n=1}^\infty 1/f(n)$ is convergent ?
I know that $\exists M>0$ such that $|f(x)|< Mx, \forall x\in ...
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1answer
50 views
Uniform Continuity in Topological Spaces?
Is it already in literature this generalized notion of uniform continuity in an arbitrary topological space (not necessarily in exactly the same form)? Let $(X, T_{1})$, $(Y, T_{2})$ be topological ...
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1answer
37 views
If $g:X \to [0,\infty)$ is defined by $g(x)=d(x,f(x))$, prove that $g$ is uniformly continuous on $X$.
Let $f:X \to X$ be a function on a compact metric space $(X,d)$ such that
$d(f(x),f(y)) < d(x,y)$ for all $x\neq y$
a) If $g:X \to [0,\infty)$ is defined by $g(x)=d(x,f(x))$, prove that $g$ is ...
1
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2answers
28 views
Uniform continuity on an open interval?
Suppose I want to check if $f(x)$ is uniform continuous on a bounded interval $I$ (for eg open interval $(0,1)$), given that it is continuous on $I$. How do I do that?
My approach: Take $\bar{I}$, ...
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3answers
123 views
Bounded Harmonic Functions on the Disk
Denote by $\mathbb{D}$ the open unit disk in $\mathbb{R}^2$.
Is it possible to find a bounded harmonic function $u : \mathbb{D} \to \mathbb{R}$ that is not uniformly continuous?
I tried using ...
3
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3answers
210 views
Real Analysis - Continuity
a. Give an example of a function defined everywhere on the interval $[0,1]$, which does not achieve its maximum.
b. Give an example of a function defined on $\mathbb{R}$, that is nowhere continuous.
...
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3answers
85 views
How to show that if $f^2(x)$ is uniformly continous function then f is uniformly continous
$f:R\to [0,\infty)$ is function such that $f^2(x)$ is uniformly continuous on R then I have to show that f is uniformly continuous ?
My attempt :
$|f^2(x)-f^2(y)|<\epsilon$ for $|x-y|<\delta$
...
2
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3answers
42 views
If $f(x)$ is a continously differentiable function such that $f'(x)$ is unbounded, then $f(x)$ is not unifromly continuous?
Suppose $f(x)$ is a continuously differentiable function on $\mathbb{R}$ such that its derivative is unbounded.
Claim: $f(x)$ is not uniformly continuous.
Assume f(x) is uniformly continuous, then ...
1
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0answers
37 views
Approximating a continuous function by a sawtooth function
I need to understand a proof in the book "Lecture notes on topology and geometry" by Singer and Thorpe, from 2.5 Applications
Theorem 1: There exists a continuous real-valued function $f\in C([0,1])$ ...
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2answers
41 views
Is the function uniformly continuous? .
$f_n(x)= \dfrac{nx}{1+n^4x^4}$ , $x∈\mathbb{R}$
I fixed $\varepsilon >0$, looking for $\delta >0$, so for every $x, y ∈ \mho$ with $|x-y|<\delta$ it is valid $|f(x)-f(y)|<\epsilon$.
So: $...
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1answer
64 views
If $\,f$ and $g$ are continuous with compact support then $f*g$ (convolution ) is also continuous with compact support [closed]
I do not know how to find a compact that satisfies $f$ and $g$ support.
Could someone explain how find this new compact ?
How $f$ and $g$ are continuous in a compact then are bounded in this ...
2
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3answers
78 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
38 views
Is $ f \colon (1, +\infty) \to \mathbb{R} , f(x) = \sin \frac{1}{x} $ uniformly continuous? [duplicate]
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
vote
1answer
14 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.
0
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1answer
13 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 $\delta$
2
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1answer
41 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
68 views
Is $f(x) = \frac{1}{x}$ uniformly continuous on $(1, \infty)$? [closed]
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
48 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}...
