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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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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2answers
42 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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3answers
47 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
25 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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2answers
71 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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2answers
58 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
64 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 ...
4
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1answer
59 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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0answers
22 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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0answers
47 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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2answers
28 views

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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1answer
43 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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2answers
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
42 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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1answer
39 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 ...
3
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2answers
169 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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1answer
46 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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2answers
31 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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3answers
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 $f:X\to \mathbb{R}$. ...
4
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3answers
95 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
67 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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29 views

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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2answers
28 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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0answers
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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1answer
69 views

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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1answer
34 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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147 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
41 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, ...
3
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2answers
58 views

Continuity vs. Mapping open sets to open sets?

I have a question and I have no idea how to solve this: One problem in my Real Analysis text book says: Show that if $\ell$ is a nonzero linear functional on a normed vector space not necessarily ...
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1answer
49 views

inverse of uniformly continuous function is uniformly continuous?

Inverse of uniformly continuous function is uniformly continuous? Assume that $ X,Y$ are metric spaces and let $f:X\to Y$ such that $f$ is bijective and ...
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2answers
48 views

Axiom of choice : continuous function and uniformly continuous

How I proof that every continuous function f in [0,1] is uniformly continuous, without axiom o choice? I took this from the book Axiom of Choice from Horst Herrilich He had a observation that ...
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1answer
48 views

If $f: (a, b) \to \Bbb R$ is uniformly continuous, then $g: [a, b] \to \Bbb R$ is continuous

Fact 1: If $A$ is a subset of a metric space, $Y$ is a metric space, and $f: A \to Y$ is uniformly continuous, then if $\{x_n\}$ is Cauchy in $A$, then $\{f(x_n)\}$ is Cauchy in $Y$. Fact 2: ...
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1answer
27 views

Uniform continuity preserves uniqueness of convergent sequences?

If $f: (a, b) \to \Bbb R$ is uniformly continuous, $\{x_n\}$ and $\{x'_n\}$ are sequences in $(a, b)$ with $x_n \to b$, $x'_n \to b$, $f(x_n) \to y$, and $f(x'_n) \to \overline{y}$, prove $y = y'$. ...
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36 views

Proving continuity using Weierstrass M-test

I am asked to use the Weierstrass M-test to show that the following function is continuous on $A = \mathbb{R}\setminus \mathbb{Z}$ $$f(x) = \sum_{n=1}^\infty \frac{1}{x+n} + \frac{1}{x-n}$$ My ...
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Is $\frac{1}{x}$ on [0,$\infty$] continuous at zero?

Taking the definition of continuity, two of three conditions are met, i.e. a) We would have to define $f(0)=\infty$, but normally division by zero is not well-defined. b) The limit $\lim_{x\to ...
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Why did mathematicians introduce the concept of uniform continuity?

I have solved many problems regarding uniform continuity, but still I can't understand the following: Is there any practical application of this concept, or it is just a theoretical concept? Is there ...
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0answers
48 views

Is this function bounded and what about uniform continuity?

There is a function $ \phi (x): \mathbb{R}^n \longmapsto \mathbb{R}^n$ which satisfies conditions (1) and (2). (1): $\langle\phi(x_1)-\phi(x_2),x_1-x_2\rangle \leq \gamma_1 \| x_1-x_2 \|^2$ (2): $ ...
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56 views

Show $f(x) = x\sin{x}$ is not uniformly continuous on $\mathbb{R}$ [duplicate]

I know it's not Lipschitz continuous because its derivative is unbounded but I'm not sure if not Lipschitz continuous implies not uniformly continuous. Thanks
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1answer
38 views

Difference between continuous and uniformly continuous functions on a dense metric subspace.

Let $X$ be a dense subset of metric space $(\tilde X,d)$. Let be $(Y,d')$ be a complete metric space and $ f: X \rightarrow Y$ a continuous mapping. It follows from density that for all points in ...
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80 views

Is $\sqrt{x}$ uniformly continous in $\mathbb{R}^+$?

We are given this function: $f:R^+\rightarrow R,x\rightarrow \sqrt{x}$. We need to prove that this function is uniformly continuous. My proof is this one but i'm not sure is it complete and right. ...
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1answer
51 views

Holder continuous functions embedded in Sobolev

For simplicity, I will consider $u\in W^{1,p}(\Omega)$, where $1<p<\infty$ and $\Omega\subset\mathbb{R}$ is open and bounded. I am able to show that \begin{equation} |u(x)-u(y)| \leq ...
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What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference via geometrically? What is the best way to describe the difference between these two concepts to someone else? Where the ...
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0answers
14 views

continuity and convergence [duplicate]

If we have a continuous function that converges on a compact subset of a metric space does it imply that it converges uniformly in general, or is this only in the case if f is monotonic (Dini's ...
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1answer
45 views

Proving that the characteristic function is uniformly continous.

I am trying to prove that the characteristic function is uniformly continuous. I understand how to get to this bound: And I would like to find the $\delta$ as a function of $\epsilon$ but I am ...
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2answers
73 views

“$\sigma$-uniform continuity”

Let $X$ be an arbitrary metric space and $f:X\to\mathbb R$ a bounded continuous function. Is it possible to choose a countable sequence $(A_n)_{n\in\mathbb N}$ of (preferably open or closed) subsets ...