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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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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 $V\in \mathcal{U}$, $(f\times f)^{...
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$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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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$\frac{1}{x}$ 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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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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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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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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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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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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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 ...
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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 $f|_{L} : L\to\mathbb{R}$ ...
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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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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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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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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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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; ...
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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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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}\...
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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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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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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 ...
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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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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 ...
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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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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 $|x-y|<\delta$ ...
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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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.
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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 $\delta$
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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.
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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 ...
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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}...