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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bijective uniformly continuous function from a subset of the Cantor set $K$ onto $X.$

Suppose $(X, d)$ is a non-empty metric space. Then $X$ is totally bounded if, and only if, there exists a bijective uniformly continuous function from a subset of the Cantor set $K$ onto $X.$ Proof: ...
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1answer
40 views

Fundamental solution Laplace-Poisson equation

Let $\Phi:\mathbb{R}^n\setminus\{0\}\rightarrow\mathbb{R}$ be the fundamental solution of the Laplace equation (see e.g. in the book of Evans). For a function $f\in\mathcal{C}_c^2(\mathbb{R}^n)$ we ...
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40 views

To show that $f(x)= \frac{1}{x}$ is not uniformly continuous on $(0,\infty)$

To show that $f(x)= \frac{1}{x}$ is not uniformly continuous on $(0,\infty)$ ATTEMPT Choose $\epsilon$ to be less than 1 Let $x_1=\delta$ and $x_2=\frac{\delta}{2}$ Then $|x_1-x_2|$ is less than ...
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Is $f(x) = x^3 \sin \frac{1}{x} $ uniformly continuous on $(0, \infty)$?

Since the derivitive of $f$ is bounded on a neighborhood of $0$, $f$ is uniformly continuous on $(0, M)$ where $M$ is any positive number. I'd like to prove that $f$ is uniformly continuous on a ...
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Given any non compact set $A \subset \mathbb R^n$ does there exist a continuous function $f: A \to \mathbb R$ which is not uniformly continuous?

It's well known that if $ A \subset \mathbb R^n$ is compact then every continuous function $f:A \to \mathbb R$ is uniformly continuous.So the obvious question is: Given a non compact set $A \subset ...
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27 views

Uniformly $\beta$-continuous functions (jumps no greater than $\beta$) converge uniformly to $f$, is $f$ continuous?

Let $(X,\rho)$ be a compact metric space, we say a function on $X$ is uniformly $\beta$-continuous if, for every $\epsilon > 0$, there exists $\delta > 0$ such that if $\rho(x,y) < \delta$, ...
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55 views

uniform continuity only using theorems

$f : [0,\infty) \to \mathbb{R}$ be continuous on $[0,\infty)$ and $\lim\limits_{x \to \infty} f(x)=0$. Prove that f is uniformly continuous on $[0,\infty)$ . Can this be proved by using the theorems ...
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1answer
44 views

Geometry of Analysis

I am a recently graduated student and doing Post Graduation now. I often come across uniform convergence, uniform continuity etc. As we all know that we check continuity and convergence easily by just ...
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50 views

Uniform continuity of a bounded function on an unbounded interval

Are all bounded continuous functions on an unbounded interval uniformly continuous ?
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36 views

Uniform Continuity of $\frac {1}{x}$ on [$a, \infty$) for positive $a$

$\frac {1}{x}$ behaves nicely in that it's monotone and the derivative is monotone also. So on [$a,\infty$] it can be seen that the $\delta$ which will work everywhere is the $\delta_1$ at the end ...
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29 views

Uniform continuity on union of sets

If a function $f(x)$ is uniformly continuous on two completely separated intervals $[a,b]$ and $[c,d]$, then is it true that $f(x)$ is uniformly continuous on $[a,b]\cup [c,d]$? I also think that ...
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16 views

Analysis Proof- different conditions.

A continuous function on $[a,b]$ is also uniformly continuous on $[a,b]$. The following tries to illustrate what happens when the interval is not closed: Show: $f(x) = \frac{1}{x} $ is not ...
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2answers
73 views

let $f$ be continuous on $[a,b]$. Prove that $f$ is integrable on $[a,b]$ [closed]

I know I will need to use the definition of uniform continuity but I don't know how to go about proving $f$ is integrable
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1answer
74 views

Prove that function is not uniformly continuous

I have to prove that the function $$f(x)=\left\{ \begin{array} \\ x^2 \cos\frac{1}{x},& x\neq 0\\ 0,& x=0 \end{array} \right.$$ is not uniformly continuous on $\mathbb{R}$ and I just can't ...
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17 views

given f uniformly continuous function in an open subset, does a continuous function g exist in the closed subset and f(x)=g(x)

Let $f : \Bbb {R}^n \to \Bbb{R}$ be a uniformly continuous function in the open subset $B(0,1) = \{ x \in \Bbb{R}^n : \sum_{i=1}^n x_i^2 < 1 \}$ does a continious function g which is defined in $ ...
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2answers
31 views

Prove uniformly continuity at $\infty$ to continuous function

Say $f:[0,\infty) \to \mathbb{R}$ is a continuous function. Assume $\lim_{x \to \infty}[f(x)-ax]=b$ for some $a,b \in \mathbb{R}$ and prove $f$ is uniformly continuous in $[0, \infty)$ So if the ...
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22 views

Continuous function's property proof using slightly different epsilon/delta definition.

I was asked to prove/disprove the following: If $f:\mathbb{R} \to \mathbb{R}$ continuous function, so for all $x \in \mathbb{R}$ and $\epsilon > 0$, exists $\delta > 0$ s.t. if $y \in ...
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1answer
30 views

Uniform Continuity implies Continuity

Let $f$ be a function from a metric space $X$ to a metric space $Y$. Show that if $f$ is uniformly continuous on $X$ then $f$ is continuous on $X$. Show that the converse is not true. Uniform ...
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48 views

What does Dini continuity mean?

What does Dini continuity (the integral condition) mean visually? Description of Dini contuity: https://en.wikipedia.org/wiki/Dini_continuity
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66 views

How to show $\sqrt{|x|}$ is not Lipschitz continuous?

$f(x) = \sqrt{|x|}$ is a famous example of a function which is not Lipschitz continuous but is uniformly continuous. This link shows detailed explanation of it. Here provides the figure of this ...
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1answer
41 views

“continuously differentiable $\subseteq$ Lipshitz continuous” with $f(x) = x^2$

In the Wiki, it says: continuously differentiable (i.e. class $C^1$) $\subseteq$ Lipshitz continuous. Consider the simplest example ($x,y\in \mathbb{R}$): $$f(x) = x^2$$ It is not Lipshitz ...
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1answer
26 views

Exploit uniform continuity

Suppose that $f:X \rightarrow \mathbb{R}$ is uniformly continuous, where $X$ is a compact metric space. By definition of uniform continuity, there exists $\delta$, independent of point chosen such ...
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78 views

Why do we care if a function is uniformly continuous? [duplicate]

There are a lot of question regarding whether a function is or is not uniformly continuous or just continuous and there are a lot of $\epsilon_s$ and $\delta_s$ trying to show whether a function is ...
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60 views

Why is $f(x) = x^2$ uniformly continuous on [0,1] but not $\mathbb{R}$

According to How exactly can't $\delta$ depend on $x$ in the definition of uniform continuity? There is a lot of agreement that $x^2$ is not uniformly continuous. But is $x^2$ uniformly ...
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3answers
70 views

Proving $f(x)=1/x$ on $(0,1 )$ is not uniformly continuous

My questions are about the reasoning made in the note http://folk.uib.no/st00895/MAT112-V12/unif-kont.pdf (which is in Norwegian). To prove that $f(x)=\frac{1}{x}$ is not uniformly continuous, the ...
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1answer
61 views

Continuous and bounded imply uniform continuity?

I am thinking about this since couple hour. Is a continuous and bounded function $f:\mathbb R\to\mathbb R$ uniform continuous too? I didn't found a counter example and thus I tried to prove this like ...
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23 views

topology of uniform convergence on compacts and strong operator topology

I am trying to understand the proof for some lemma in a book, but the part the authors label as trivial is not trivial to me at all. Any help is appreciated. Here is the trivial part of the lemma: ...
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49 views

Is this a valid proof of $f:[a,b]\to\mathbb{R}$ continuous $\implies$ $f$ uniformly continuous?

I'm wondering whether this proof (which I tried to put together myself, for practice) is valid, or if you can find any flaw in my reasoning, or if I'm doing something unnecessarily complicated. ...
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Is the continuous extension theorem true when the range space of $f$ is not complete?

So the problem is Exercise $13$, Chap. $4$ of Principles of Mathematical Analysis by Rudin: Problem Let $E$ be a dense subset of metric space $X$, and let $f$ be a uniformly continuous real function ...
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45 views

If $f:R\to R$ is uniformly continuous so $ \lim_{x \to \infty} f(\sqrt {x^2+5})-f(x) = 0$?

Prove that if $f:R\to R$ is uniformly continuous so $$ \lim_{x \to \infty} f(\sqrt {x^2+5})-f(x) = 0$$ any ideas? thanks
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21 views

Bounding $\delta$ without the loss of generality while proving non uniform continuity

To show $f(x)$ isn't uniformly continuous on an interval $I$, I show that there is an $\epsilon$ such that for every $\delta$ there exist $x, y \in I$ such that $|x-y|<\delta$ and ...
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3answers
44 views

Can we determine uniform continuity from graphs?

Can we know from the graph of a function that whether the function is uniformly continuous or not? If the set on which the function is defined is not compact, so what one can say in that case?
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42 views

Uniform continuity.

Check if the mappings $\mathbb{R}\to\mathbb{R},x\mapsto x^2$ and $[0,\infty[:\mathbb{R},x\mapsto \sqrt{x}$ are uniformly continuous. I was going through some old exams our teacher ...
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174 views

Show that $f(x) = x^2$ is not uniformly continuous on $[0,\infty)$

Ok, I know the same question has already been asked here, and I am not looking for an answer even though my proof looks kind of the same. But, I need to know whether or not I am on the right track. ...
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47 views

Please could someone check my proof that continuous implies locally Lipschitz

I have produced a false proof but can't spot the mistake. I proved the following (false) statement: Let $U \subseteq \mathbb R^n$ be open and $f: U \to \mathbb R^n$ be continuous. Then $f$ is ...
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33 views

Why is this function uniformly continuous?

Assuming $X$ is a normed space. Why is a function $f:K\rightarrow\mathbb{R}$ uniformly continuous on a subspace $K\subset X$, if $K$ is sequentially compact and $f$ is continuous?
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60 views

Is the map $x\mapsto x^2+x$ uniformly continuous on $(0,\infty)$?

I want to find whether $x\mapsto x^2 + x$ is uniformly continuous on $(0,\infty)$, I know that $x\mapsto x^2$ is not uniformly continuous on this interval, however am having difficulty grasping ...
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1answer
52 views

If $(f_n)_{n\geq 1} \to f$ uniformly and $(g_n)_{n\geq 1} \to g$ uniformly then $(f_n \circ g_n)_{n\geq 1} \to f\circ g$ uniformly

EDIT: Let $X$ be a compact metric space, $A$ a set. Let $(f_n)_{n\geq 1}$ be a sequence of continuous functions from $X$ to $\mathbb{R}$, and suppose $f_n \to f$ uniformly where $f:X\to \mathbb{R}$. ...
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3answers
100 views

Bolzano -Weierstrass Theorem and uniform Continuity

The following problem has hints, but I am unable presently to use it. Suppose $f$ is uniformly continuous on $(a,b]$, and let $\{x_n\}$ be any fixed sequence in $(a,b]$ converging to $a$. Show ...
3
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1answer
70 views

Finding Function's Extension and Its Unique Existence.

Let $$A= \left\{\frac j{2^n}\in [0,1] \mid n = 1,2,3,\ldots,\;j=0,1,2,\ldots,2^n\right\} $$ and let $$ f:A\rightarrow R $$ satisfy the following condition: There is a sequence $ \epsilon_n \gt 0 $ ...
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Let $ f:I \to \mathbb{R} , I=(0,1) $ be uniformly continuous. Then exists $ \lim_{n\to\infty} f(\frac{1}{n}) $

True. Since $f$ is continuous (because all uniformly continuous function is continuous), we can assume: $$ f\left(\lim_{n\to\infty} \frac{1}{n}\right) $$ Since $ \lim_{n\to\infty} \frac{1}{n} $ is ...
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Uniform continuity on a dense subset implies uniform continuity on the set.

Let $f: (M_1,d_1)\to (M_2,d_2)$ be continous. If $f$ is uniformly continous on a dense subset $V$ of $M_1$, show $f$ is uniformly continous on $M_1$. Could someone provide some hints? (no full ...
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30 views

Continuity and Uniform Continuity on half closed intervals

I have been stuck on the following problem for a long time : Prove that if a function $f:(a,b]\to\mathbb R$ is continuous, then it is uniformly continuous if and only if $\lim_{x\to a^+}f(x)$ ...
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44 views

Uniformly continuous functions on the interval [duplicate]

Let $f:[1,\infty)\to\mathbb R$ be uniformly continuous. Prove $\exists$ $M > 0$ s.t $$\frac{\big|f(x)\big|}{x} \leq M, \hspace{11pt} \forall x\in[1,\infty)$$
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Hard problem about Lipschitz condition. This is a very unusual case. [closed]

Can you please help me with proving Lipschitz condition? Let: $f\colon\mathbb{R}\longrightarrow \mathbb{R}$ satisfy $\lvert f(x)-f(y) \rvert\leq C\lvert x-y\rvert^\alpha$ for some constants $C\gt 0$, ...
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36 views

How can I formally write $f(x) \to 0$ when $x \to \infty$

I've just proven that if $f:\mathbb{R} \to \mathbb{R}$ is uniformly continuous in $[a,b]$ and it is also uniformly continuous in $[b,+\infty)$ then $f$ is uniformly continuous in $\mathbb{R}_{\geq ...
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1answer
63 views

Show $\sum e^{-nx + \cos(nx)}$ is defined on $(a, \infty) $ for any $a>0 \dots$

I want to prove that $\sum e^{-nx + \cos(nx)}$ is defined and continuous on the given interval of $(a, \infty)$ where $a >0$. Then, how exactly do I show it is defined? It just seems trivial, ...
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166 views

If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?

Let $X$ be a metric space such that every continuous function $f:X \to \mathbb R$ is uniformly continuous ( here $\mathbb R$ is equipped with the standard euclidean metric ) , then is it true that for ...
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1answer
35 views

differentiable and uniform continuity of f and F

Given $f: \Bbb R \to \Bbb R$. define new function: $F(x) =\frac{f(x)-f(a)}{x-a}$ for $x\neq a$. Prove that $f$ is differentiable at $a$ if and only if $F$ is uniformly continuous in some punctured ...
3
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1answer
42 views

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent:

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent: a) $f$ is uniformly continuous in $X$. b)For every pair of sequences $(x_n), (y_n) \subseteq X$ such that $ d(x_n, ...