# 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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### On uniformly Lipschitz function.

Let $f:[t_{0}, T]\times X\to X$ be uniformly Lipschitz in $X$, what does uniformly Lipschitz function mean for this case? This question arises from the Theorem 1.7 which appears in the image. Thanks ...
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### Sequential Criterion for the absence of Uniform Continuityy

I am reading Abbott's Understanding Analysis wherein the sequential criterion for the absence of uniform continuity is given as follows: A function $f : A \to\mathbb R$ fails to be uniformly ...
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### continous functions on compact metric spaces are uniformly continuous [closed]

While reading about continuous functions on compact metric spaces , we study a theorem that a continuous function on a compact metric space is uniformly continuous. While going through the proof after ...
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### Groups where every continuous function is uniformly continuous

Let $G$ be a topological group, and $(X, d)$ a metric space. The function $f : G \to X$ is left uniformly continuous if for all $\varepsilon > 0$ there exists an identity neighbourhood $U$ such ...
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### Proof that a continuous function on a compact set is uniformly continuous, compact set not containing infimum

I'm reviewing the proof of the following statement (Baby Rudin 4.19): Let $f$ be a continuous mapping of a compact metric space $X$ into a metric space $Y$. Then $f$ is uniformly continuous on $X$. ...
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### Proving that $f(x) = \frac{1}{x}$ is not uniformly continuous over $(0,1)$ - Approach to choosing the correct $x$ and $y$

I've looked at the various other posts on this question and I've also worked on uniform continuity in the past. But right now as I work through Spivak's Calculus, I'm seeing that I haven't actually ...
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### Help me in understanding Heine-Borel theorem's proof

I am trying to understand the proof of Heine-Borel theorem given here. If I understand the meaning behind the phrases in box, I think it is enough for me to understand the proof. Please explain the ...
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### Uniform continuity of a function: multidimensional distribution function

If a distribution function $F$ defined on $\mathbb{R},$ is continuous then it's uniformly continuous, this is easy to prove, since more generally if $f$ is continuous on $\mathbb{R}$ and has finite ...
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### $f:\mathbb{R} \to \mathbb{R}$ is differentiable function such that $f'$ is bounded .Is $f+f'$ uniform continuous?? [closed]

$f:\mathbb{R} \to \mathbb{R}$ is differentiable function such that $f'$ is bounded, then we know $f$ is uniform continuous. But is $f+f'$ uniform continuous? I think no but I am not able to find a ...
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### Check function uniform continuity [closed]

The task is to check function uniform continuity in terms of the following set $\{(x,y): x^2+y^2 \geq 2\}$: $$f(x,y)=(x^2+y^2)\cdot \sin\left(\frac{1}{x^2+y^2}\right)$$ Can you help me with this one? ...
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### Show that the inequality $|f(x)-f(y)| \leq M||x-y||+\epsilon$.

Suppose $K \subset \mathbb{R}^n$ is a compact set and $f:K \rightarrow \mathbb{R}$ is continuous. Let $\epsilon >0$ be given. Prove that there exists a positive number $M$ such that for all $x$ and ...
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### uniform continuity on disjoint intervals

Consider the following statements (a) If $f$ is uniformly continuous on disjoint closed intervals $I1,I2,......,In$, then $f$ is uniformly continuous on $\cup_{j=1}^n Ij$ (b) If $f$ is uniformly ...
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### Value of a function that is uniformly convergent.

"Let $f$ be a function of $\mathbb{R}$ into $\mathbb{R}$ such that $\vert f(x)-f(y) \vert\leq\frac{\pi}{2}\vert x-y\vert^2$ for all $x,y\in\mathbb{R}$, and such that $f(0)=0$. What is $f(\pi)$?". ...
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### Does this phenomenon regarding volumes of images of small balls hold uniformly?

Let $\Omega \subseteq \mathbb{R}^n$ be a nice domain with smooth boundary (say a ball), and let $f:\Omega \to \mathbb{R}^n$ be smooth. Set $\Omega_0=\{ x \in \Omega \, | \, \det df_x =0 \}$ In this ...
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### Example of the two uniformly equivalent metrics, one is bounded while another is not.

Can anyone gives me an example such that two metric space (X,d1), (X,d2) are uniformly equivalent. And the metric space (X,d1) is bounded and metric space (X,d2) is not bounded?
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### Some questions about the proof of the properties of the uniform equivalent.

I want to prove some properties of uniformly equivalent metrics. Suppose (X,d) and (X,p) are uniformly equivalent, then the identity map and its inverse are uniformly continuous. The former is ...
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### How to prove the uniform equivalence is indeed an equivalence relation on the class of metrics on X

May I ask a homework question? I'm just wandering the equivalence relation is defined on two sets while the uniformly equivalent is defined on two metrics. How can they be equal? And how to prove that?...
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### Continuity implies uniform continuity

What shown below is a reference from "Analysis on manifolds" by James R. Munkres First of all I desire discuss the compactness of $\Delta$: infact strangerly I proved the compactness of $\Delta$ in ...
### $f>0$ uniformly continuous and $\int_0^{\infty} f(x) \,dx \leq M$ imply $\lim_{x \to \infty} f(x)=0$?
I know that if $f$ uniformly continuous and $\int_0^{\infty} f(x) \,dx = c$, then $\lim_{x \to \infty} f(x)=0$ (link: $f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply \$\lim_{x \...