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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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Prove that if $f$ is continuous, and has two asymptotes, then it is uniformly continuous (Argument check)

The exercise is the following: Let $f:\mathbb{R} \rightarrow \mathbb{R} $ continuous such that $\lim_{x\rightarrow+\infty} f(x) = \ell_1$ and $\lim_{x\rightarrow-\infty} f(x) = \ell_2$ for some ...
Fausto Martinez's user avatar
-2 votes
1 answer
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How can I prove that if $f:[0,+\infty)\to[0,1]$ is continuous and increasing, then it is uniformly continuous? [closed]

The problem is the following: Let $f:[0,+\infty)\rightarrow\mathbb{R}$ such that it is continuous, strictly growing and its image is contained in $[0,1]$, is uniformly continuous. I've tried in ...
Fausto Martinez's user avatar
-5 votes
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27 views

Show the range of $p$, where $f(x) = x^p \sin(1/x)$ is continuous. [closed]

Defined $f$ as, $$ \begin{align} f(x)= \begin{cases} x^p \sin(\frac{1}{x})\,\, &(x>0)\\ 0 &(x=0).\end{cases}\end{align}$$ Then, show the range of $p$ where, (1)$f$ is continuous function, ...
brian's user avatar
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Uniform continuity inequality check [closed]

This question is about how the person in a linked question (below), managed to derive a certain inequality. I present the linked question as well as my own derivation. I believe my question is a ...
nz_'s user avatar
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1 answer
57 views

Uniform continuity of $f(x) = \cos(e^x)$

We have a theorem for Uniform continuous functions A function $f : A \to\mathbb R$ is uniformly continuous on $A$ if and only if for all sequences ($x_n$) and ($y_n$) in A such that when $\lim_{x\to\...
Ak9848's user avatar
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Uniform convergence of sequence of continuous functions implies uniform continuity?

I recently encountered a question where I'm not sure if I answered it correctly, given that I didn't use one of the assumptions. So here's the question: Prove that if a sequence of continuous ...
Darrell Tan's user avatar
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1 answer
42 views

Convergence of series using uniform continuity

Let $f:(0,+\infty)\rightarrow R$ be uniformly continuous. Prove that infinite series : $\sum\frac{1}{n}(f(n)-2f(n+1)+f(n+2))$ is convergent. I saw some behaviour of this sum, and that for some first ...
AveriX's user avatar
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0 answers
35 views

A continuous extension of an uniformly continuos function on an interval

Let $a, b \in \mathbb{R}$ such that $a<b$ and let $f:(a,b)\to\mathbb{R}$ be a continuous function. Prove that are equivalent i) There exists a continuous function $g:[a,b]\to\mathbb{R}$ such that $...
MrGran's user avatar
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A converse to Heine's Theorem about uniformly continuous functions on the real line

Let $I$ be a non-trivial interval such that every continuous function on $I$ is uniformly continuous. Prove that $I$ is closed and bounded (that is, compact). My solution: suppose, for the sake of ...
MrGran's user avatar
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A clarification regarding the definition of uniform continuity of a function defined in a subset of $\mathbb R.$

Let $f:A\to \Bbb R$ where $A\subseteq \Bbb R$. We say that, $f$ is uniformly continuous on $A$ if for any $\epsilon\gt 0$ there exists $\delta(\epsilon)=\delta\gt 0$ such that for any $x_1,x_2\in A$ ...
Thomas Finley's user avatar
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19 views

Relation between local Lipschitz continuity constant and Lipschitz smoothness constant of a function.

Using local Lipschitz continuity of $f(\cdot)$: \begin{align} f(\mathbf{a}) &\leq f(\mathbf{b})+ L_0 \lVert{\mathbf{a}-\mathbf{b}}\rVert \end{align} In the FedProx Paper (https://arxiv.org/pdf/...
Nazreen Shah's user avatar
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1 answer
39 views

Quantifiers uniform continuity

According to this answer: https://math.stackexchange.com/a/2582334/1098426 We know $\forall x \ \exists y \ \forall z$ differs from $\forall x \forall z\ \exists y$ insofar as $y$ depends on $x$ ...
isaac's user avatar
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Complex analysis Bruce P. Palka- Ex 5.35 [closed]

In complex analysis: Let $S = \{ z : z = 0 \text{ or } |\text{Arg } z| \leq \alpha \}$, where $0 < \alpha < \pi$. Verify that the function $f(z) = \sqrt{z}$ is uniformly continuous on $S$, ...
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Show that $f$ is uniformly continuous if limit on $f(x)+f'(x)$ at infinity is finite [duplicate]

Let $f:[0,\infty)\to\mathbb{R}$ be continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ suppose: $$\lim_{x\to\infty}{f(x)+f'(x)}=5$$ Show that $f$ is uniformly continuous. I've thought of ...
pompabalompa's user avatar
2 votes
1 answer
75 views

What properties of metric spaces are not preserved by uniformly continuous isomorphism?

Compactness and connectedness are preserved by homeomorphism, in the sense that if two metric spaces $(X,d_X)$ and $(Y,d_Y)$ are homeomorphic and $(X,d_X)$ is compact then it follows that $(Y,d_Y)$ is ...
Gimmel_007's user avatar
2 votes
1 answer
17 views

Characteristic functions: upper bound on |phi(t)e^(-itx)|

This post about characteristic functions asks about absolutely integrable $\phi(t)$. This answer mentions "we can find $R$ such that $\int_{\mathbb R\setminus[-R,R]}\lvert\phi(t)\rvert dt<2\pi\...
johnsmith's user avatar
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If $f,g$ are uniformly continuous and $f$ is bounded and non-periodic, then $fg$ is not necessarily uniformly continuos

I've just begun my grad program and we were introduced to this problem in our Analysis I course: consider two uniformly continuous functions $f$ and $g$, say from $\mathbb R$ to $\mathbb R$, where $f$ ...
Arthur's user avatar
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Is ${\log(x)}^2$ uniformly continuous on $(1, \infty)$?

Is $f(x)={\log(x)}^2$ uniformly continuous on $(1,\infty)$? I have found the above question in an old exam and was wondering how to solve it. Actually, a few friends and I have found 2 different ...
Keroten's user avatar
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0 answers
38 views

verify the function is lipschitz or not

Question $a>0$ ,$b>0$, $x_0\in \mathbb{R}$,$t_0\in \mathbb{R}$ . The set D is defined as $D=\{(t,x)\in \mathbb{R^2}:|t-t_0|\leq a , |x-x_0|\leq b \}$ . $a(t,x)$ is a continuous function on $D$ . ...
RRR's user avatar
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uniform continuity of $f(x) = \frac{x}{1 +x^2}$ on $\mathbb{R}$

$f(x) = \frac{x}{1 +x^2}$ on $\mathbb{R}$ to comment about its uniform continuity, I tried to figure out the definition " A function $f(x)$ is said to be uniformly continuous on a set $S$, if for ...
Svidi Runs's user avatar
2 votes
1 answer
53 views

About a step in the proof of "A linear transformation between two normed spaces is uniformly continuous if it is bounded."

Theorem: A linear transformation between two normed spaces is uniformly continuous if it's bounded. Proof: Let $X_1$ and $X_2$ be normed linear spaces with norms $\| \cdot \|_i,i=1,2$ and let $T$ be a ...
Jonathen's user avatar
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1 vote
1 answer
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Does there exist topology $T$ on the real numbers such that the class of $T$-$T$-continuous and uniformly continuous functions coincide?

Does there exists a topology $T$ on the set of real numbers $\mathbb{R}$ such that the $T$-$T$-continuous functions from $\mathbb{R}$ to $\mathbb{R}$ are precisely the uniformly continuous functions? ...
user107952's user avatar
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1 answer
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Whether a function is continuous or not by following conditions. [closed]

If there exists a unique function $$f : R → R$$ such that $f$ is continuous at $x = 0$, and such that for all $x ∈ R$ such that $$f(x) + f(\frac{x}{2}) = x$$ Then Will it be continuous$?$ I tried many ...
Ayush Kumar Singh's user avatar
2 votes
1 answer
154 views

$L^2$ convergence implies expected value of $f(X_n)$ converges to expected value of $f(X)$

This is from a second undergraduate course in probability. Show if $X_n \overset{L^2}{\rightarrow} X$, $f$ a uniformly continuous and bounded function, then $\mathbb{E}f(X_n) \rightarrow \mathbb{E}f(X)...
johnsmith's user avatar
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3 votes
2 answers
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Why uniform continuity does not imply equicontinuity?

Rudin's book "Principles of Mathematical Analysis" gives the following definition for equicontinuity: A family $\mathscr{F}$ of complex functions $f$ defined on a set $E$ in a metric space $...
MC2's user avatar
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4 votes
1 answer
143 views

Prove that if characteristic function is absolutely integrable then corresponding density function will be uniformly continuous.

So the point is that you obtain density function through the characteristic function with the inversion formulas. That's the part of the task. Then it supposed to be uniformly continuous. That is, if $...
Egor's user avatar
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5 votes
0 answers
162 views

Finding the limit of an integral using the Stone-Weierstrass theorem

I found the following problem while preparing for some math olympiad in my country. Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
Shthephathord23's user avatar
-1 votes
1 answer
52 views

If a function is bounded and continuous does it imply it is Lipshitz function [closed]

Let f be a real valued function. If f is bounded and continuous on an Interval does it imply it is Lipshitz function on that interval? Also does there exist an unbounded function on (0,1) which is ...
Shash's user avatar
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1 vote
2 answers
97 views

Prove that $f(x)=x^2$ is not uniformly continuous on $[a,\infty).$

Prove that $f(x)=x^2$ is not uniformly continuous on $[a,\infty).$ The solution given is as follows: Let us choose $\epsilon > 0.$ Then for any two points $x,c\in [a,b]$ satisfying $|x-c|< \...
Thomas Finley's user avatar
0 votes
0 answers
46 views

Proof uniform continuity of Lipschitz functions

I am stuck on a problem about uniform continuity of Lipschitz functions. A function $F:\mathbb{R}\to\mathbb{R}$ is called Lipschitz continuous if there exists a $L\in\mathbb{R}_{>0}$​ such that for ...
ArtanR's user avatar
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0 answers
102 views

If $f''$ is uniformly continuous and $f$ converges to $0$ then $f'$ converges to $0$

Let $f : (0, +\infty) \rightarrow \Bbb R$, $f''$ is uniformly continuous and $f$ converges to $0$ as $x$ approaches $\infty$. Show that $f'$ converges to $0$ as $x$ approaches $\infty$. To be honest, ...
ABlack's user avatar
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1 vote
1 answer
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About uniform continuity of $y = \sqrt{x}$ and general intuition on uniform continuity

When I check uniform continuity of a (continuous) function, I generally look for a part where the slope diverges to infinity (e.g. $y = x^2$, $y = \frac{1}{x}$). For $f(x)= \sqrt{x}$ for $ x\in [0, \...
hnbm's user avatar
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0 answers
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Uniform continuity of $h$ satisfying $ h(a,y) \geq \alpha$ for al $y$ and $a$ is fixed

Let $h:\mathbb{R}^d \times \mathbb{R}^n \to \mathbb{R}$ be a $C^1$ mapping and $a\in \mathbb{R}^d$. Assume that there is $\alpha>0$ such that $$ h(a,y) \geq \alpha,$$ for all $y\in \mathbb{R}^n$. ...
hanava331's user avatar
1 vote
0 answers
45 views

Uniform continuity and the order of quantifiers

I’m taking my first course in real analysis, and I’m trying to prove the following proposition. Proposition: If $f:S\to\mathbb{R}$ is uniformly continuous, then $f$ is continuous. In comparing ...
Jaebeom Yim's user avatar
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1 answer
49 views

Disproving uniform continuity of a function using Cauchy continuity

From my understanding of uniformly continuous functions, they will, by definition, map Cauchy sequences to Cauchy sequences (thus preserving the Cauchy sequence in its transformation). If a function ...
aort01's user avatar
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1 vote
1 answer
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Proof that if $f$ is uniformly continuous then for every Cauchy sequence $(x_n)$ with $a < x_n < b$ $f(x_n)$ is also cauchy.

I need to show that the following statement holds true: Given $a, b \in \mathbb{R}$, $a < b$, $f: (a, b) \to \mathbb{R}$, $f$ continuous. Show that $f$ uniformly continuous $\Rightarrow$ $\forall$ ...
Felix Gervasi's user avatar
0 votes
1 answer
83 views

Absolute continuous implies uniform continuous?

Let $f:[a,b]\to \mathbb{R}^n$ be an absolutely continuous function. Is it uniformly continuous? I know that if $n=1$ it is true and it's called Heine's theorem, but what about $n\geq 1$?
BrianTag's user avatar
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4 votes
1 answer
92 views

Are uniformly continuous functions on $(a,b)$ also uniformly continous on $[a,b]$?

I did my research on this question and I found two answers that I feel contradict one another: this first one says it's not true and this second one says it is. When I asked my professor about it he ...
Thomas Grenier's user avatar
0 votes
2 answers
100 views

Confusion about bounded derivative implies uniform continuity

I'm currently learning about uniform continuity and was introduced to an intuitive way of "seeing" uniform continuity by observing the derivative of the function across an interval. Notably, ...
aort01's user avatar
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0 answers
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Continuous function on closed interval is uniformly continuous...

I am not getting the statement: continous function on closed interval is uniformly continuous... what I know $f(x)=\sin 1/x$ is continuous...and not uniformly continuous...but If I consider closed ...
math student's user avatar
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1 vote
2 answers
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Uniform Continuity between $f$ and $|f|$

$f$ is a continuous function on $\mathbb{R}$,if $|f|$ is uniformly continuous on $\mathbb{R}$, then how about the uniform continuity of $f$? I know that if $f$ is uniformly continuous, then $|f|$ is ...
Hance Wu's user avatar
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1 vote
1 answer
64 views

Stuck while proving the function $g:A=[0,\infty)\to \Bbb R$ such that $g(x)=\sqrt x$ is uniformly continuous on $A.$ [duplicate]

Prove that the function $g:A=[0,\infty)\to \Bbb R$ such that $g(x)=\sqrt x$ is uniformly continuous on $A.$ My attempt so far: Let $I=[0,2]$. If we restrict $g$ to $I$ then by uniform continuity ...
Thomas Finley's user avatar
2 votes
1 answer
155 views

If a function has a uniform Lipschitz constant on an open cover of $\mathbb{R}$, is it globally Lipschitz?

I had a question regarding locally and globally Lipschitz functions. Suppose there exist some real $\delta > 0$ and $M > 0$ such that $f : \mathbb{R} \to \mathbb{R}$ satisfies $|f(x) - f(y)| \...
MathNeophyte's user avatar
1 vote
2 answers
98 views

Continuous function $f$ is uniformly continuous on $D$ $\iff$ (when $x_n,y_n \in D$ then $|x_n-y_n| \rightarrow 0 \implies |f(x)-f(y)| \rightarrow 0)$

I need to understand a way to prove this statement $f$ is uniformly continuous on $D$ $\iff$ (when $x,y \in D$ then $|x-y| \rightarrow 0 \implies |f(x)-f(y)| \rightarrow 0)$ I know $\delta$ can depend ...
John Doe's user avatar
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1 vote
0 answers
19 views

is there any specific way or algorithm to verify if a function is uniformly continuous or not?

I want to know like for continuity of a function at a given point we can check the limit of the function at that point and the value of the function at that point but how to verify if a function is ...
Flash 's user avatar
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1 vote
0 answers
110 views

Prove that there exists $M>0$ such that for every $x\geq 1$ it satisfies $|f(x)|\leq Mx$

Let $f:\mathbb R\rightarrow \mathbb R$ be a uniformly continuous function in the interval $[1,\infty)=I$. Prove that there exists $M>0$ such that for every $x\geq 1$ it satisfies $|f(x)|\leq Mx$. ...
Chess player's user avatar
1 vote
1 answer
77 views

Sequential definition of uniform continuity

The equivalence of the “$\varepsilon-\delta$” definition of continuity and the sequential definition is often used, but I’m wondering if there is possibly a sequential definition of uniform continuity?...
Nick's user avatar
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1 answer
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Why $\delta$ is choosen between $0$ & $1$ in showing $f(z) =\frac{1}{z}$ is not uniformly continuous in the region $|z|<1$

In the below attached picture, there is, solution provided for showing $f(z) =\frac{1}{z}$ is not uniformly continuous un the region $|z|<1$ By definition of uniform continuity, we have for given ...
General Mathematics's user avatar
5 votes
0 answers
109 views

Continuous extension of a function which retains the modulus of continuity

Question Let $S \subset \mathbb R^d$ be compact and $f : S \to \mathbb R$ be continuous (hence uniformly continuous) on $S$. A function $\omega : (0,\infty) \to (0,\infty)$ is called a modulus of ...
Sarvesh Ravichandran Iyer's user avatar
0 votes
1 answer
27 views

On the relation between simple and uniform continuity

In the Real and Functional Analysis course, we were introduced to the concept of uniform continuity (UC) without much elaboration. I am currently exploring the nuances of uniform continuity compared ...
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