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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Showing a measurable function is uniformly continuous

Let $E \subset \mathbb{R}$ be a measurable set with $m(E) < \infty$. Define the functoin $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = m(E \cap (-\infty,x])$. Prove that $f$ is uniformly continuous on $...
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87 views

Uniform continuity general case

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function defined on (all of) the real axis. Assume that $$f(x) \rightarrow a, \quad \text{for } x \rightarrow -\infty$$ $$f(x) \rightarrow b, ...
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25 views

Two continuous functions on closed interval and smallest element

Assume $f,g\colon [0,1]\to [0,1]$ are continuous functions and $$ g(0)<f(0) \ \ \text{and}\ \ g(1)>f(1)$$Is it true that there exists smallest $t\in(0,1)$ such that $$ g(t)\geq f(t)$$ I know ...
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57 views

Uniform continuous with simple properties

$f$ is a map from $f:\mathbb{R} \rightarrow \mathbb{R}$. Let $f$ be continous and assume $f$ is both uniformly contiuous on $(-\infty,0]$ and $[0,\infty)$ . Show that $f$ is uniformly continuous on ...
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24 views

a guess relation betwenn series and infinite integral convergence

Background: Our real analysis textbook says if function f is a non-negative decreasing function on range $[1\,+\infty)$, thus $\int_{1}^{+\infty}f(x)\,dx$ converges is equivalent to $\sum_{n=1}^{\...
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13 views

Semiflow and intersection of open sets on a compact subset of topological semigroup

Let $X$ be a Urysohn space and $T$ be a topologically semigroup. Also let $\varphi:T\times X\to X$ be a continuous action. Consider a compact set $K\subseteq T$. Q1. Let $\{V_t\}_{t\in K}$ be a family ...
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3answers
62 views

Is $x\sin\left(\frac{1}{x}\right)$ uniformly continuous in $\mathbb{R}$? [duplicate]

Let $f:\mathbb{R}\xrightarrow{}\mathbb{R}$ with rule defined as $$ f(x) = \left\{\begin{array} xx\operatorname{sin}\left(\frac{1}{x}\right) &\text{ if }x\neq0 \\ 0 &\...
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1answer
26 views

Analysis - Prove a function that maps the unit Euclidean ball to R with bounded partial derivatives is uniformly continuous

I am stuck with this problem from my textbook and I cannot see the solution. I'm certain the solution is fairly simple and I am just missing the mark somehow. Any help would be appreciated. Suppose ...
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21 views

Showing this multivariable function is uniformly continuous

Working on a problem from my class's textbook, and I'm stuck. The problem is seemingly easy, and I'm fairly certain I should be using the $\epsilon$-$\delta$ definition of uniform continuity, but I ...
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2answers
46 views

Confused about something that has to do with uniform continuity

I am confused about something that has to do with uniform continuity. The rectangles in the image above represent the regions formed by the following intervals, by the definition of uniform continuity ...
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1answer
70 views

Show $f(x)=x^TBx$ is not uniformly continuous if $B$ is symmetric

I am in the initial fase of learning analysis. I am working on the following problem: Consider the quadratic form $f \, : \, \mathbb{R}^k \to \mathbb{R} $ given by $f(x)=x^TBx$ for $x \in \mathbb{R^k}$...
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1answer
23 views

Converse of Heine-Cantor Theorem

Heine-Cantor theorem asserts that every continuous function on a compact metric space is uniformly continuous. The converse is , if every continous function is uniformly continuous on a metric space X ...
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3answers
120 views

Uniformly continuous function on bounded open interval is bounded

Let $f(x)$ be uniformly continuous on a bounded open interval $a<x<b$. Show that $f$ is bounded (i.e. $\exists M$ such that $|f(x)|\le M \ \forall x\in (a,b)$). To be honest, I have no idea how ...
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43 views

If $(f_n)$ is an orthonormal basis and $\lambda_n\to\infty$, is $T(t)x:=\sum_ne^{-λ_nt}\langle x,f_n\rangle f_n$ uniformly continuous?

Let $H$ be a complex Hilbert space, $(f_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $(\lambda_n)_{n\in\mathbb N}\subseteq(0,\infty)$ be nondecreasing with $\lambda_n\xrightarrow{n\to\infty}\...
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1answer
28 views

Prove that a function is uniformly continuous.

A function defined on an interval I is said to be uniformly continuous on I if to each $\epsilon$ there exists a $\delta$ such that $|f(x_1)-f(x_2)| < \epsilon$, for arbitrary points $x_1, x_2$ of ...
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2answers
36 views

Physical Interpretation of uniformly continuous function.

I have been reading material on uniformly continuous functions. And going through the problems where we have to prove that a function is not uniformly continuous or otherwise. A function defined on an ...
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2answers
48 views

Proving $\cosh(\sqrt{1+x^2})$ is not uniformly continuous in $\mathbb{R}$

Given $f(x) = \cosh(\sqrt{1+x^2})$ I am trying to show that $f(x)$ is not uniformly continuous. Specifically: $\exists\varepsilon\ \forall\delta\ \exists x,y \in \mathbb{R}: \ |x-y| < \delta \wedge ...
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1answer
20 views

If $\tau$ is a vector topology on a normed space s.t. a closed ball is $\tau$-compact, is a continuous function on a closed ball even uniformly cont.?

Let $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$, $r>0$, $X$ be a normed $\mathbb K$-vector space, $\tau$ be a vector topology on $X$ s.t. $\overline B_r^X(0):=\{x\in X:\left\|x\right\|_X\le r\}$ ...
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1answer
25 views

Uniform continuous function is bounded by a linear function.

I was solving the next exercise: Let $f:\mathbb{R}\to\mathbb{R}$ be an uniformly continuous function. Prove that there exist $a,b\in\mathbb{R}^{+}$ such that, for all $x\in\mathbb{R}$, $|f(x)|\leq a|...
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61 views

Is a function that is continuous and doesn't blow up always uniformly continuous?

A function $f: D \rightarrow \mathbb{R}$ is called uniformly continuous if and only if $ \forall \varepsilon>0 \exists \delta>0 \forall x, x_{0} \in D:\left|x-x_{0}\right|<\delta \Rightarrow\...
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16 views

To prove non uniform continuity of a composite function

We have to discuss the uniform continuity of the function $$ f(x)= \tan(\frac{\pi}{2}\cos(\frac{x}{2}))$$ where f:$(0,\pi)\to \Bbb R$ I know we can use the fact that the function's continuity cannot ...
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33 views

How to show function is uniformly continous on certain interval

Let $ f\colon \mathbb{R} \to \mathbb{R}$ be $$x \mapsto \begin{cases} \frac{(x-1)^2}{x^2+1}, &|x| \neq 1 \\2, &|x|=1\end{cases} $$ How can I show that $f$ is uniformly continuous on $[-2,2]\...
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52 views

Is the function $f(z) = z$ continuous on $|z| = 1$? [closed]

Is the function $f(z) = z$ continuous on $|z| = 1$? Here $z$ is a complex number. I'm scratching my head here as $z_{0}$ is a modulus. How do I calculate this?
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58 views

$f: (a, b] \to \mathbb{R}$ continuous. Prove: $f$ continuously resumable in $a \iff f$ is uniformly continuous

Let $f: (a, b] \to \mathbb{R}$. Prove that: $f$ is continuously resumable in $a \iff f$ is uniformly continuous. My approach: "$\implies$:" Since $a$ is continuously resumable, then $a$ has ...
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1answer
27 views

Difference in calculating continuity and uniform continuity

I've done some exercises on continuity using the $\varepsilon$-$\delta$ definition. Now going through my script I saw that there is a stricter version of continuity called uniform continuity. After ...
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41 views

Induced representations: space of continuous functions on $G$ to a Hilbert space

EDIT: Answered in MathOverflow @https://mathoverflow.net/questions/382324/induced-representations-space-of-continuous-functions-on-g-to-a-hilbert-space Let $G$ be a locally compact group, $H$ a ...
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41 views

Are the integrals uniformly continuous and differentiable on $\mathbb{R}$?

I want to check if one of the following is uniformly continuous and differentiable on $\mathbb{R}$. $\int_0^{x^3}\sin t\, dt$ $\int_0^x\cos (t^3)\, dt$ $\int_0^x[t]\, dt$ $\int_0^xe^{t^3}\, dt$ To ...
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1answer
16 views

Uniform continuity choosing delta problem

I am trying to show that $x^2$ is uniformly continuous on the set union of intervals [n,n+$n^{-2}$] for all positive integer n. I have also checked the previous posts. But my question is that for a ...
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2answers
38 views

A counter example on uniform continuity

Let $f:(0,1]\rightarrow \mathbb{R} $ be a continuous function and bounded is it uniformly continuous? I know this isn't true, but I can't find a good counter example I was thinking something like ...
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61 views

Every uniformly continuous function $\mathbb Q\to \mathbb Z$ is constant.

First of all, is continuity not enough? Secondly, my attempt is since it is uniform continuous. For $\epsilon=1/2$, for every ball on $\mathbb Z$ i.e. $B(n,\epsilon)\subset \mathbb Z$, with $B(n,\...
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1answer
45 views

Prove that $f(x)=e^{\frac{1}{x}}$ is uniformly continuous.

Problem: Prove that $f(x)=e^{\frac{1}{x}}$ is uniformly continuous on $(a,1)$ for $a \in (0,1)$. My solution: By using the Lagrange theorem for $x,y \in (a,1)$ we have $$\frac{f(x)-f(y)}{x-y}=f'(c)=e^{...
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41 views

What properties does this function have?

Let $f\colon\mathbb{R} \rightarrow \mathbb{R}$ be an $\textbf{uniformly continuous}$ function. Then by definition $\forall\epsilon>0 \quad \exists\delta>0\quad$such that $\forall x,y\in\mathbb{...
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A standard name for an “inversely (uniformly) continuous” map

Is there a standard name for a map $f: X → Y$ between metric spaces such that for every $ε > 0$ there is $δ > 0$ such that $d(x, y) < δ$ if $d(f(x), f(y)) < ε$ for $x, y ∈ X$ and/or for a ...
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1answer
34 views

Is this argument for uniform continuity good?

Consider the function $f:(-\infty, 1)\to \mathbb{R}$, $f(x)=\frac{1}{x-1}$. I would like to prove that this function is uniformly continuous. My argument is as follows: the function is continous ...
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25 views

Show that f : ]0;1[ $\longrightarrow \mathbb{R} $, uniformly continuous on ]0;1[ is bounded [duplicate]

Let f : ]0;1[ $\longrightarrow \mathbb{R} $, uniformly continuous on ]0;1[. I want to show that f is bounded. My idea is to show that $lim_{x \rightarrow 0+} f(x)$ and $lim_{x \rightarrow 1-} f(x)$ ...
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44 views

Uniformly continuous $f$ such that $\lim_{n \to +\infty}f(nx) = 0 \implies \lim_{x \to +\infty} f(x)= 0$

Let $f : \mathbb R \to \mathbb R$ a uniformly continuous function such that $\forall x > 0$ $$\lim_{n \to +\infty}f(nx) = 0$$Show that $$\lim_{x \to +\infty} f(x)= 0$$ I cannot use Baire Category ...
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1answer
23 views

Prove or disprove: function is uniformly continuous

Let $f$ be a function on $\mathbb R$. If $$|f(x)-f(y)|<4|x-y|$$ for all real numbers $x$ and $y$, then $f$ is uniformly continuous on $\mathbb R$. My answer is as follows: The definition of ...
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183 views

How to prove this problem about uniform continuity

If $f$ Continuous bounded on $\mathbb{R}$,and $$\lim_{h\to0}\sup_{x\in\mathbb{R}}\left|f(x+h)-2f(x)+f(x-h)\right|=0.$$ prove that $f$ uniformly continuous on $\mathbb{R}$. I try to find a $\delta>0$...
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1answer
34 views

Non-uniformly continous function

I have found a topic here (https://www.sciencedirect.com/topics/mathematics/uniformly-continuous-function) in Theorem 3.3.10 that: There is a continuous function $f$ on $[0, 1]$ which is unbounded, ...
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1answer
44 views

Bounding uniformly continuous functions [closed]

Show that if $f \colon \mathbb{R} \to \mathbb{R}$ is uniformly continuous, then there is an $L > 0$ such that $$|f(x)| \leq L \cdot (1 + |x|).$$ I'm quite stuck here. Any hints would be much ...
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1answer
62 views

Let $g(x)=\sin(x)$ and $f(x)=x$,show that the function $fg$ defined by $(fg)(x)=f(x)g(x)$ is not uniformly continuous on $\mathbb R$.

Let $g(x)=\sin(x)$ and $f(x)=x$,show that the function $fg$ defined by $(fg)(x)=f(x)g(x)$ is not uniformly continuous on $\mathbb R$. A function $f:D \to \mathbb R$ is not uniformly continuous on $D$ ...
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1answer
30 views

Show that the function $f:[a,\infty] \to \mathbb R$ defined by $x \mapsto \frac{1}{x}$ for $a$ a positive number is uniformly continuous.

Show that the function $f:[a,\infty] \to \mathbb R$ defined by $x \mapsto \frac{1}{x}$ for $a$ a positive number is uniformly continuous. The function is continuous if and only if $$\forall \epsilon&...
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1answer
81 views

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

I want to know if this proof is rigorous enough. ...
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1answer
59 views

Solution check: Uniform continuity

While reading Bartle's book, i came across the following: Problem: Prove that if $f$ is uniformly continuous on a bounded subset $A$ of $\mathbb{R},$ then $f$ is bounded on $A$. Attempt: Given $A \...
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23 views

uniformly continuous prove question

My proof for question 6: Can anyone help me verify my proof for question 6 and help me prove question 7? I really have no ideas on question 7, Thank you!
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29 views

Proving Heine criterion for multivariable uniform continuity in $f:\mathbb R^2 \rightarrow \mathbb R$

I am trying to show that for $f: D \rightarrow \mathbb R$, where $D \subset \mathbb R^2$, $f$ is uniformly continuous in $D$ if and only if for every $\{(x_n, y_n)\}_{n=1}^\infty, \{(w_n, z_n)\}_{n=1}^...
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1answer
42 views

Arbitrarily close uniformly continuous functions

While preparing for my Real Analysis exam I tried to solve the following problem I found: Problem: Let $A \subseteq \mathbb{R}$ and suppose that $f: A \rightarrow \mathbb{R}$ has the following ...
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2answers
30 views

Trying to prove uniform continuity for a function

I have been trying to prove that the following function is not uniformly continuous: Consider $A=(0,\infty), B=(-\infty,0),$ and $f: A \cup B \rightarrow \mathbb{R}$ given by $$ f(x)=\left\{\begin{...
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3answers
35 views

Uniform continuity regarding union of open sets

I am stuck at the following Real Analysis problem from my book: Problem: Give an example of two open sets $A$ and $B$ and a continuous function $f: A \cup B \rightarrow \mathbb{R}$ such that $f\mid A$ ...
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25 views

What is the definition of asymptotically flat convergence?

I'm reading an article about a sliding modes control approach which mentions that a given function Ln(cosh(*)) is nearly linear outside the small neighborhood of the origin but partial asymptotically ...

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