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".

Continuity of a function is purely a local property, whereas uniform continuity is a global property that applies over the whole space.

Definition:

A real-valued function $~f~$ defined on a set $~E~$ of real numbers is said to be uniformly continuous provided for each $\epsilon~\gt ~0~$, there is a $~\delta~\gt~0~$ such that for all $~x,~x'~\in~E~$,

if$~\quad |x-x'|\lt \delta~$, then $\quad |f(x)-f(x')|\lt~\epsilon~$.

${}$

A mapping from a metric space $~(X,\rho)~$ to a metric space $~(Y,\rho)~$ is said to be uniformly continuous, provided for every $\epsilon~\gt ~0~$, there is a $~\delta~\gt~0~$ such that for all $~u,~v~\in~E~$,

if$~\quad \rho(u,~v)\lt \delta~$, then $\quad \sigma (f(x)-f(x'))\lt~\epsilon~$.

  • uniform continuous $\implies$ continuous, but converse is not true.

e.g., The function $~f(x) = \frac{1}{x}~$ is continuous on $~(0,1)~$ but not uniformly continuous.

  • The $~\delta~$ here depends on $~\epsilon~$ and on $~f~$ but that it is entirely independent of the points $~x~$ and $~y~$. In this way, uniform continuity is stronger than continuity and so it follows immediately that every uniformly continuous function is continuous.

Reference:

https://en.wikipedia.org/wiki/Uniform_continuity#Generalization_to_uniform_spaces

1946 questions
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Continuous mapping on a compact metric space is uniformly continuous

I am struggling with this question: Prove or give a counterexample: If $f : X \to Y$ is a continuous mapping from a compact metric space $X$, then $f$ is uniformly continuous on $X$. Thanks for your help in advance.
81
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3 answers

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 any wide application of this concept in any…
79
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6 answers

What is the intuition behind uniform continuity?

There’s another post asking for the motivation behind uniform continuity. I’m not a huge fan of it since the top-rated comment spoke about local and global interactions of information, and frankly I just did not get it. Playing with the definition,…
48
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9 answers

A game with $\delta$, $\epsilon$ and uniform continuity.

UPDATE: Bounty awarded, but it is still shady about what f) is. In Makarov's Selected Problems in Real Analysis there's this challenging problem: Describe the set of functions $f: \mathbb R \rightarrow \mathbb R$ having the following properties…
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How does the existence of a limit imply that a function is uniformly continuous

I am working on a homework problem from Avner Friedman's Advanced Calculus (#1 page 68) which asks Suppose that $f(x)$ is a continuous function on the interval $[0,\infty)$. Prove that if $\lim_{x\to\infty} f(x)$ exists (as a real number), then…
Jubbles
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Prove that the function$\ f(x)=\sin(x^2)$ is not uniformly continuous on the domain $\mathbb{R}$.

If I want to prove that the function$\ f(x)=\sin(x^2)$ is not uniformly continuous on the domain $\mathbb{R}$, I need to show that: $\exists\varepsilon>0$ $\forall\delta>0$ $\exists{x,y}\in\mathbb{R}\ : |x-y|<\delta$ and $|\sin(x^2) -…
Jeroen
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What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference geometrically? What is the best way to describe the difference between these two concepts? Where the motivation of Uniform Continuity came from? Thank You.
Bumblebee
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$\sqrt x$ is uniformly continuous

Prove that the function $\sqrt x$ is uniformly continuous on $\{x\in \mathbb{R} | x \ge 0\}$. To show uniformly continuity I must show for a given $\epsilon > 0$ there exists a $\delta>0$ such that for all $x_1, x_2 \in \mathbb{R}$ we have $|x_1 -…
user104235
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Uniform continuity, uniform convergence, and translation

Let $f:\mathbb R \to \mathbb R$ be a continuous function. Define $f_n:\mathbb R \to \mathbb R$ by $$ f_n(x) := f(x+1/n). $$ Suppose that $(f_n)_{n=1}^\infty$ converges uniformly to $f$. Does it follow that $f$ is uniformly continuous? Note: the…
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How do I show that all continuous periodic functions are bounded and uniform continuous?

A function $f:\mathbb{R}\to \mathbb{R}$ is periodic if there exits $p>0$ such that $f(x+P)=f(x)$ for all $x\in \mathbb{R}$. Show that every continuous periodic function is bounded and uniformly continuous. For boundedness, I first tried to show…
28
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2 answers

Is the distance function in a metric space (uniformly) continuous?

Let $(X, d)$ be a metric space. Is the function $x\mapsto d(x, z)$ continuous? Is it uniformly continuous?
Renato
25
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Prove that a function whose derivative is bounded is uniformly continuous.

Suppose that $f$ is a real-valued function on $\Bbb R$ whose derivative exists at each point and is bounded. Prove that $f$ is uniformly continuous.
TurboMan
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Connected unbounded sets $S\subset \Bbb{R}^n$ such that $x\mapsto ||x||^t$ is uniformly continuous on $S$?

Spending the night perusing my old answers, and this question left me wondering about the following. Let's equip $\Bbb{R}^n$ with the usual Euclidean metric, and let us consider the map $N_t:\Bbb{R}^n\to \Bbb{R}$, $N_t(\vec{x})=||\vec{x}||^t$. The…
Jyrki Lahtonen
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composition of two uniformly continuous functions.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ are two uniform continuous functions. Which of the following options are correct and why? $f(g(x))$ is uniformly continuous. $f(g(x))$ is continuous but not…
24
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Is a bounded and continuous function uniformly continuous?

$f\colon(-1,1)\rightarrow \mathbb{R}$ is bounded and continuous does it mean that $f$ is uniformly continuous? Well, $f(x)=x\sin(1/x)$ does the job for counterexample? Please help!
Marso
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