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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Criterion for uniform continuity multivariable calculus

Let $D\subset \Bbb{R}^n$ be an open set (non empty) and convex. Let $f:D\to \Bbb{R}$ be a $C^1(D)$ function s.t $\exists C \in \Bbb{R}$ s.t $\|\nabla f(x)\|\leq C \ \forall x \in D$. Show that $f$ is ...
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Why does the interval $[0, ∞)$ defined as a confine interval?

I dont understand that $[0, ∞)$ is a confine intervall. As x approaches ∞ is has a endpoint, or is it something that I am missing? Does it mean that $(-∞,∞)$ is also a confine intervall? Thanks for ...
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How to prove that if $f$ is Lipschitz of order $\alpha>0$ on an interval, then $f$ is uniformly continuous on this interval?

Spivak's Calculus, chapter 11 "Significance of the Derivative", problem 37: A function $f$ is Lipschitz of order $\alpha$ at $x$ if there is a constant $C$ such that $$(*)\ \ \ \ \ |f(x)-f(...
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The space of Lipschitz continuous functions is dense in that of uniformly continuous functions?

Let $(X,d)$ be a metric space. Then The space of bounded uniformly continuous functions is dense in that of bounded continuous functions w.r.t. the supremum norm. ref The space of bounded Lipschitz ...
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Uniformly continuous function implies almost Lipschitz continous.

I wrote a proof of the following exercise but I am dubious there may be an incorrect step in it. I would appreciate your comments: If $f:\mathbb{R}^n\to\mathbb{R}^m$ is uniformly continuous on $D\...
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Cannot understand this proof of uniform continuity.

I dont get the last step of the proof that explains that Let $f:(a,b)\rightarrow \mathbb{R}\ $is continuous. $f\ $uniformly continuous if $\ \lim _{x\to a^+}\left(f\left(x\right)\right)\ $ and $\ \lim ...
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Proof of $1/x^2$ being uniformly continuous on $[a,\infty)$ for $a>0$

When I try to prove this I find that $$|f(x)-f(y)|=|\frac{1}{x^2}-\frac{1}{y^2}|<|\frac{1}{a^2}|+|\frac{1}{a^2}|=\frac{2}{a^2} $$ I can't see how to proceed by here, how can I get $|f(x)-f(y)|<\...
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For any non-empty subset $A$ of real numbers define $f_A : \mathbb{R} \to \mathbb{R}$ by, $f_A(x)= \inf \{ |x-y| : y \in A \}$

For any non-empty subset $A$ of real numbers define $f_A : \mathbb{R} \to \mathbb{R}$ by, $$f_A(x)= \inf \{ |x-y| : y \in A \}$$ Let, $A,B$ be disjoint non-empty sets of real numbers such that $f_A(x)...
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Proving $1/x$ is uniformly continuous on $[1/2,1]$

We wish to prove that $f(x)=\frac{1}{x}$ defined on $[1/2,1]$ is uniformly continuous. Scratch work: Take $x,y \in [1/2,1]$, $|f(y)-f(x)|=|\frac{1}{y}-\frac{1}{x}|=|\frac{y-x}{xy}|$ At most $x=1,y=1$ ...
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is $e^x$ uniformly continuous on the open $(0,\infty)?$ [closed]

I tried coming up with two sequences by the non-uniform criterion such that $F(X_n)-F(Y_n)\neq0$ but I had no luck in that. How do I show its uniformly continuous using the definition?
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Uniformly continuous map that separates $2$ closed subsets with positive distance of a metric space.

I'm trying to build a continuous map that separates $2$ disjoint closed subsets of a metric space. Moreover, if the distance between them is positive, then the map is uniformly continuous. Let $(E, d)...
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Is $f(x) = \frac{1}{x^2}$ uniformly continuous on $(0,1]$ or $[1, \infty)$ or both?

Here is the question I am trying to answer: Are the following functions uniformly continuous? Prove it. $(a)$ $f: (0, 1] \to \mathbb R$ with $f(x) = \frac{1}{x^2}.$ $(b)$ $f: [1, \infty) \to \mathbb R$...
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Show that the function $f(x)=x^3, (x \in \Bbb{R} )$ is not uniformly continuous?

I have been assigned this question but the solution given uses a different method so i don't know if my solution is correct $$\exists \bar{\epsilon} \gt0 \quad \text{s.t.} \quad|\bar{x}-\bar{y}|\lt\...
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Showing that the polygonal function $f(t) = a_p + n\left(t - \frac{p - 1}{n}\right)(a_p - a_{p-1})$ is uniformly continuous on the interval $[0,1]$

Let $a_0,\dots,a_n \in \mathbb{R}$ be fixed and $f$ be defined as $f(t) = a_p + n\left(t - \frac{p - 1}{n}\right)(a_p - a_{p-1})$ for $\frac{p - 1}{n} \leq t \leq \frac{p}{n}, 1\leq p \leq n$. I'm ...
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Proof of uniform continuity of $f(x) = \inf\{\mid x - a \mid : a \in S\}, S \subset \mathbb{R}$

I was wondering if my proof of the uniform continuity of $f(x) = \inf\{| x - a | \colon a \in S\}, S \subset \mathbb{R}$ was correct. $ | f(x) - f(y) | = | \inf\{|(x-a)|\} - \inf\{|(y-a)|\} | \leq | \...
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How to prove function $d_S(x)$ is uniformly continuous

Suppose $S\subset \mathbb R$ is a arbitrary subset and function $d_S(x):\mathbb R\to \mathbb R$ $$d_S(x)=\inf\{|x-a|:a\in S\} $$ prove this function is uniformly continuous Since I know the ...
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Does every uniformly continuous function admit a continuous and concave modulus of continuity

Let $f:(X,d)\rightarrow (Y,\rho)$ be a uniformly continuous function and suppose that $\omega$ is a modulus of continuity for $f$. Can we always find an laxer concave and continuous modulus for $f$ ...
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Find a continuous bounded function $f:(0,1]\to \mathbb{R}$ that is not uniformly continuous.

Find a continuous bounded function $f:(0,1]\to \mathbb{R}$ that is not uniformly continuous. Extend $f$ with continuity in such a way that $f(0)=0$ and find the oscillation $\omega _f(0)$ of $f$ at $0$...
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Uniformly continuous integral example [closed]

Can someone, please, help me with an example of improper integral of some function $$\int_{0}^{+\infty} f(x, y)dx,$$ $c \leq y \leq d$, which is uniformly continuous on $c \leq y \leq d$, but for ...
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About the product of two uniformly continuous functions

I am interested in the following question: Assume $f,g: [0,+\infty)\to\mathbf R$ are both uniformly continuous, and $g(x)\to 0$ as $x\to+\infty$. Can this guarantee that $h(x)\equiv f(x)g(x)$ is ...
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Proving $f:[-1,1] \to \mathbb{R}$ with $f(x)=x^3$ is uniformly continuous

Can anyone verify the steps in my proof are correct? Scratch work: $|f(x)-f(y)| =|x^3-y^3|=|x-y||x^2+xy+y^2|$ (at this point note that we already know $|x-y|<\delta$ for some $\delta>0$) $$|x-y||...
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Prove the function $1/z$ is uniformly continuous on $\frac{1}{3}\le\left|z\right|<1$,is the function uniformly continuous on $0<\left|z\right|\le1$?

Prove that the function $1/z$ is uniformly continuous on $\frac{1}{3}\le\left|z\right|<1$, is the function uniformly continuous on $0<\left|z\right|\le1$? My definition of uniform continuity is:...
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every uniformly continuous semigroup of operators can be extended to a uniformly continuous group

I'm reading Vrabie and I'm in trouble with this remark. Given a uniformly continuous semigroup $\{S(t)\mid t\geq 0\}$, since $S(t)$ is invertible and can be extended to a uniformly continuous group ...
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Uniformly continuous semigroups are analytic

I know that every analytic $C_0$-semigroup is differentiable and the every differentiable semigroup is norm continuous. I wanted to know where uniform continuity fits in the above picture. My ...
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Bounded linear functional on a subspace of a Hilbert space implies uniformly continuous?

If $f: M \to \mathbb{R}$ is a bounded linear functional where $M$ is a subspace of a Hilbert space, then is it the case that $f$ is uniformly continuous? I definitely believe that $f$ is continuous ...
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Can I find a modulus of continuity like this?

Let $W \colon [0,\infty) \to [0,\infty) $ and suppose that I have the following estimate: for every $\delta >0$ $$W(\theta)\leq \delta + C_{1/ \delta} \theta^2$$ for every $\theta>0$ where $C_{1/...
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2 answers
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Show that F is uniformly continuous function where $F(x_{1},x_{2},\dots, x_{n})= \max\{ |x_{1}|,|x_{2}|,\dots, |x_{n}| \}$ .

$F\colon \mathbb R^n \to \mathbb R$ defined as $F(x_{1},x_{2},\dots, x_{n})= \max\{ |x_{1}|,|x_{2}|,\dots, |x_{n}| \}$ Show that F is uniformly continuous function. So we have to show $\forall \...
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Proof of uniformly continuous on infinite union set [closed]

Let $f:A\mapsto\mathbb{R}$ be a function. Let $A = A_1\cup A_2 \cup\ldots$ is an infinite decomposition of the domain $A$ of $f$ such that for some $\delta_0>0$ and $i\ne j\in\mathbb{N}$, we have $|...
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prove that uniform continuity on union of sets [duplicate]

Let f:A→R be a function. Let A = A1 U A2 U...U An is a finite decomposition of the domain A of f such that for some δ0>0 and i≠j in {1,2,...n}, we have |x-y| >= δ0 for all x∈Ai and y∈Aj. Show ...
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$f$ is uniformly continuous iff $ \|\tau_y f - f \|_u \rightarrow 0 $ as $y \to 0$ (Proof verification)

I'm trying to prove the following elementary fact mentioned on page 238 of Folland's Real Analysis: A function $f$ is called uniformly continuous if $\|\tau_y f - f \|_u \rightarrow 0$ as $y \to 0$. (...
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Proving the Heine-Cantor theorem on uniform continuity, but using sequences

Let $f:[a,b]\to \Bbb R$. We say that $f(x)$ is uniformly continuous if for every two sequences $x_n,y_n$ in $[a,b]$ satisfying $x_n-y_n\to 0$, we have $f(x_n)-f(y_n)\to 0$. It is well known that this ...
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Is this proof that arguing equicontinuousness in a compact set is uniformly continuous valid?

This is a theorem that was discussed in class but I came up with another proof. The theorem is: Let $X$ and $Y$ be metric spaces. If $X$ is compact, and $h$ is a set of mappings from $X$ to $Y$ and is ...
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Prove sin(x) is uniformly continuous on R with Heine-Cantor Theorem

I've already proven uniform continuity of f(x)=sin(x) on R via epsilon-delta, but I wanted to try and prove it with the Heine-Cantor-Theorem since it seems more intuitive: Now obviously with the ...
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If $\lim_{n\to \infty}f\left(\frac{1}{n}\right)=f(0)=\lim_{n\to \infty}f\left(-\frac{1}{n}\right)$ then is $f$ continuous at $x=0$

For a function $f:[-1,1]\rightarrow R$, consider the following statements: Statement 1: If $$`\lim_{n\to \infty}f\left(\frac{1}{n}\right)=f(0)=\lim_{n\to \infty}f\left(-\frac{1}{n}\right)`,$$then $f$ ...
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$A\subset A'$. Prove that $A$ is sequentially compact if and only if every continuous function on $A$ is uniformly continuous.

Suppose $X$ is a metric space, $A\subset X$ and $A\subset A'$(the set of limit points of $A$). Prove that $A$ is sequentially compact if and only if every continuous function on $A$ is uniformly ...
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Is the follwoing function an example of an uniformly continuous function?

Is $id$ an uniformly continuous function? Let $(\mathbb{N},d)$ be a metric space such that $d(x,y) = |\frac{1}{x}-\frac{1}{y}|$. Let us consider the identity map $id:(\mathbb{N},d) \to (\mathbb{N},d_{...
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Is the following function uniformly continuous on the closed and bounded interval?

Let us consider the identity function $$f:(\mathbb{R},d)\to (\mathbb{R},d_{usual}) $$ $$f:x \to x$$ Here we are considering $d(x,y)=|(x)^3-(y)^3|$ Is the function $f$ uniformly continuous on closed ...
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Uniform continuity contradiction

Question, I am learning uniform continuity and I saw a sentence which I am not sure about it ( since I can not find anything on it online ). If the derivative is not bounded, and it limit on absolute ...
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A property of a function which is continuous on a comeager set

Let $(X, d)$ be a compact metric space and $f:X\to \mathbb{R}$ is continuous on a comeager set $A\subseteq X$. Choose $p, q\in A$ with $f(p)\neq f(q)$. Is it true that there are open sets $U$ of $p$ ...
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uniform continuity - Using Lipshitz always

Question, is it possible to prove each question regarding uniform continuity with Lipshitz? I am having problem with this subject and can only answer be using Lipshitz. but I see there are some ...
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Prove that the modulus of continuity is continuous

Let $f:[a,b]\to \Bbb R$ be a uniformly continuous function and define $$ \omega(\delta) = \sup\{|f(x)-f(y)|:\ x,y\in [a,b], \ \text{ and } |x-y|<\delta\}$$ which we call the modulus of continuity. ...
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Alternative proof that a continuous function on metric spaces with compact domain is uniformly continuous

I just came up with the following proof. What I like about it is that when I came up with it I felt like I was just following my nose. It's somewhat different from the other ones I've seen, so I'd ...
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Uniform continuity of a discontinuous function over specific sets

Suppose $$f:=\begin{cases}n,\ \ \text{if}\ x\in\mathbb{N}\\0\ \ \text{otherwise}\end{cases}$$. Then, is $f$ uniform continuous over $\mathbb{N}$ or the set $A=\{n+\frac1{n}:n\in\mathbb{N}\}$? Or both? ...
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Uniform convergence of a sequence of continuous functions on a compact set

I know that the following claim is not true: if a sequence of continuous functions converging pointwise to a continuous function then the convergence is uniform. A counterexample is given here: Does ...
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$\varepsilon-\delta$ argument to show that uniform continuity of a real-valued function $f$ on some interval $A$ implies that $f$ is continuous on $A$

Here is an $\varepsilon-\delta$ argument to show that uniform continuity of a real-valued function $f$ on some interval $A$ implies that $f$ is continuous on $A$. Suppose $f$ is a real-valued ...
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A function $f(x)$ is uniformly continuous on $(a,b)$ $\iff$ it can be defined on $\{a, b\}$ such that $f$ is continuous on $[a,b]$.

A Theorem regarding Uniform Continuity states : A function $f(x)$ is uniformly continuous on $(a,b)$ $\iff$ It can be defined on the endpoints $a$, $b$ such that $f$ is continuous on the $[a,b]$. ...
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1 answer
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$f:[a,\infty) \longrightarrow \mathbb{R}$ continuous and $\lim_{x\to \infty}{f(x)}=c\in \mathbb{R}$ so $f$ is uniformly continuos in $[a,\infty)$

let be $ \ f:[a,\infty) \longrightarrow \mathbb{R} \ $. Such that f is continuous in $[a,\infty)$ and: $$\lim_{x\to \infty}{f(x)}=c\in \mathbb{R}.$$ then $f$ is uniformly continuous in $[a,\infty)$. I ...
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A bounded, continuous and monotone function must be uniformly continuous on an open interval

Consider $f : (a, b) \rightarrow \mathbb R$ for $a, b \in \mathbb R$ where $f$ is monotone, continuous and bounded. Show that $f$ is uniformly continuous. Consider a set of $k$ evenly spaced points $...
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The confusion about the proof of if function is continuous on a closed interval $[a,b]$,then it is integrable

In many proofs, it uses the fact that if a function is continuous on the interval $[a,b]$, then it is uniformly continuous on that interval Thus, statement :$\forall \epsilon>0, \exists \delta>0,...
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Wikipedia says that a function is uniformly continuous if and only if it admits a modulus of continuity.

The title statement is from wikipedia https://en.wikipedia.org/wiki/Modulus_of_continuity But this question asks about a uniformly continuous function for which there does not exist a modulus of ...
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