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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4
votes
2answers
58 views

Is $f(x)$ uniformly continuous?

$f:\mathbb R\to[0,\infty )$ is continuous such that $g(x)={(f(x))}^2$ is uniformly continuous. Then prove that $f(x)$ is uniformly continuous. I've tried to use the following method: $$ ...
0
votes
1answer
49 views

Proving $1\over x^2$ is not uniformly continuous

I need to show that $1 \over x^2$ is not uniformly continuous on the interval $(0,2]$ using the definition of uniform continuity. Definition of Uniform Continuity on a set A: Let $A \subseteq \Bbb ...
0
votes
1answer
34 views

Prove that the function $f(z) = \frac{1}{1-z}$ is not uniformly continuous on $(-1,1)$

Prove that the function $f(z) = \frac{1}{1-z}$ is not uniformly continuous on $(-1,1)$. Partial proof : Suppose $f$ is uniformly continuous. $\implies \forall \epsilon > 0, \exists \delta ...
1
vote
2answers
55 views

Show that $f$ is uniformly continuous - $\lim_{\|x\| \to \infty}f(x) = c$

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a continuous function. We suppose that there exist $c \in \mathbb{R}^m$ such that $$\lim_{\|x\| \to \infty} f(x) = c.$$ Show that $f$ is uniformly ...
1
vote
1answer
36 views

Show that $f$ is uniformly continuous - Compactness

Let $K := \{x \in C[0,1] : x(0) \in [-3,4], |x(t)-x(s)| \leq d |t^2-s^2|, \forall t,s \in C[0,1]\}$. Let $y \in C[0,1]$ and $f : K \to \mathbb{R}$ defined as $f(x)= \int_0^1 x(t)y(t)dt$. Show ...
0
votes
0answers
27 views

Uniform continuity of this function

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function. We assume that $f$ satisfies the following property: For every sequence of real numbers $(x_n)_n$, there exist a subsequence $(x_{\phi ...
0
votes
0answers
46 views

is $x\sin(\frac{1}{x^2})$ uniformly continuous on $(0,1]$?

Is $x\sin(\frac{1}{x^2})$ uniformly continuous on $(0,1]$? I am really unsure how to start this one off. I have done checks for "similar" functions like: $f=\sin(\frac{1}{x}), x\in(0,1]$, by using ...
1
vote
0answers
20 views

Uniformly continuous of a function [duplicate]

Let $f:\mathbb{R}\to \mathbb{R} $ be an uniformly continuous functions, how to prove that there are $a, b $ such that $|f(x)|\leq a|x|+b$ thanks for any suggestions
2
votes
1answer
33 views

Show $\cos(x^2)/(1+ x^2)$ is uniformly continuous on $\Bbb R$.

now here's how I did proceed. By definition a function $f: E →\Bbb R$ is uniformly continuous iff for every $ε > 0$, there is a $δ > 0$ such that $|x-a| < δ$ and $x,a$ are elements of $E$ ...
0
votes
2answers
25 views

Completeness of a certain normed space

let $(X, \| \|$) be the normed linear space of bounded uniformly continuous real valued functions defined on $R$. I need to prove that X is complete (under sup norm). My attempt: Let {$f_n$} be a ...
2
votes
2answers
54 views

Is $\frac{1}{\sin x}-\frac{1}{x}$ uniformly continuous on $(0,1)$?

So I am tasked with finding whether $\frac{1}{\sin(x)}-\frac{1}{x}$ is uniformly continuous on the open interval $I=(0,1)$. To look at the "simple" ways to prove it first: I obviously can't extend ...
0
votes
1answer
20 views

Showing uniform continuity for two intervals

Show that if $f$ is continuous on $[0, \infty)$ and uniformly continuous $[a, \infty)$ for some positive constant $a$, then $f$ is uniformly continuous on $[0, \infty)$. Here is my attempt at the ...
1
vote
0answers
10 views

Equicontinuous homotopies of families of uniformly equicontinuous functions

Let $f\colon X \to Y$ be a uniformly continuous function. Then I think it is "well-known" that it may be approximated by a Lipschitz function, and how well one can do this depends on the modulus of ...
4
votes
2answers
23 views

Showing that a function is uniformly continuous but not Lipschitz

If $g(x):= \sqrt x $ for $x \in [0,1]$, show that there does not exist a constant $K$ such that $|g(x)| \leq K|x|$ $ \forall x \in [0,1]$ Conclude that the uniformly continuous function $g$ is not a ...
2
votes
2answers
54 views

Show that $\lim_{x \to x_0} f(x)$ exist

Let $E$ a normed vector space and $F$ a Banach space. Let $A \subset E$, $x_0$ an accumulation point and $f : A \to F$ a uniformly continuous function. Show that $\lim_{x \to x_0} f(x)$ exist. ...
0
votes
2answers
27 views

Show that the function $h : A \to F \times G$ defined by $h(x) = (f(x), g(x))$ is uniformly continuous.

Let $(E,\|\cdot\|_E)$, $(F,\|\cdot\|_f)$, $(G,\|\cdot\|_G)$ three normed vector spaces. We endow the space $F \times G$ with the norm $\|(y,z)\|= \|y\|_F + \|z\|_G$. Let $A \subset E$ and let ...
2
votes
3answers
48 views

$f(x,y)=\frac{(x,y)}{\|(x,y)\|^2}$ - Show that $f$ is not uniformly continuous

Let $f: \mathbb{R^2}-\{(0,0)\} \to \mathbb{R^2}$ difined as $f(x,y)=\frac{(x,y)}{\|(x,y)\|^2}$. Show that $f$ is not uniformly continuous. I tried to solve it in using the Cauchy-Schwarz ...
0
votes
0answers
14 views

Difference between local Holder exponent and point-wise Holder exponent

What is the precise difference between local Holder exponent (resp. continuity) and point-wise Holder exponent (res. continuity)? I use the following definition for point-wise Holder continuity: ...
0
votes
0answers
18 views

For what $a$ is $t^{a}\sin \frac{1}{t}$, $t > 0$ uniformly continuous?

Without paying a whole lot of attention to what $a \in \mathbb{R}$ is, I was able to show that, since $t^{a}$ and $\sin \frac{1}{t}$ are both continuous on $(0,1]$, $x(t)$ can be extended to $y(t) = ...
0
votes
1answer
29 views

Supremum and infimum of continuous functions on $[a,b]$

Let $\{ f_\alpha (x) \}_{\alpha \in I }$ a family of continuous functions from $[a,b]$ to $\Bbb{R}$ ($ a<b $). Now we know because $[a,b]$ is compact then every $f_\alpha$ is uniformly continuous ...
0
votes
0answers
43 views

Single reference to classical results in analysis.

I am writing an expository work. And I need classical references (books or articles) that simultaneously proof the three classical results below. Any suggestion? Theorem. Let ...
0
votes
2answers
20 views

Uniformly continuous at two intervals

If I have function $f$ which is uniformly continuous at interval $[a,b)$ and at interval $[b, c)$ as well, can I conclude that $f$ is uniformly continuous at $[a, c)$? I am pretty sure that this ...
1
vote
0answers
26 views

Uniform continuity with respect to parameter.

Let $\mathbb{X},\mathbb{Y}$ and $T$ metric spaces. A family $\{f_t\}_{t\in T}$ of maps $f_t:\mathbb{X}\to\mathbb{Y}$ is uniformly continuous with respect to parameter $t$ if, $$ (\forall ...
1
vote
2answers
57 views

Proving Uniform Continuity of a function

I'm trying to prove the following theorem: Let $f(x)$ be continuous in $[0,\infty]$ and let $lim_{x \to \infty}f(x)$ be finite. Then $f(x)$ is Uniformly Continuous in $[0,\infty]$. I managed to ...
5
votes
3answers
174 views

counterexample to a “theorem” on continuity of largest deltas for continuous functions $f:[a,b]\to\mathbb{R}$

"Theorem 12" in these notes states the following (verbatim): Let $f:[a,b]\to\mathbb{R}$ be continuous and let $\epsilon>0$. For $x\in[a,b]$, let ...
1
vote
0answers
33 views

Is uniform continuity a property of the category of completely regular spaces?

If $(X, U)$ and $(Y, V)$ are uniform spaces then one has the notion of a map $f : X \to Y$ to be uniformly continuous relative to $U$ and $V$. A uniform space $(X, U)$ induces a completely regular ...
0
votes
1answer
33 views

$f(x) =\frac{1}{x^{\alpha}}$ continuity and uniform continuity

Let $f(x) =\frac{1}{x^{\alpha}}$where $x \in (0, 1)$ and $0<\alpha \le 1$. Is $f$ continuous? What about uniform continuity? Justify. I know this function is continuous in the given domain for all ...
0
votes
2answers
33 views

Proof about uniform continuity

I'm stuck on this problem: Let $C\subset \mathbb{R}^n$ be closed and unbounded. Suppose $f:C\to\mathbb{R}^m$ is continuous and such that $\lim_{x\to\infty} f(x)$ exists and is finite. ...
0
votes
3answers
33 views

Continuity and uniform continuity

I'm having trouble understanding the notion of uniform continuity, the definition states as follows: Let $f: D\to\mathbb{R}$, $f$ is uniformly continous in $X\subset D$ if: $$\forall\varepsilon >0, ...
-1
votes
3answers
34 views

If the pointwise limit is uniformly continuous, the functions in the sequence need not be so

If $f_{n}$ is a sequence of uniformly continuous functions and $f_n \to f$, then $f$ is a continuous function. Why is the converse of this statement not necessary true? Is there a simple example? ...
1
vote
3answers
75 views

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

I know that $f(x)=\frac{1}{x}$ is continuous on $(0,1)$ but why is $f(x)=\frac{1}{x}$ not uniformly continuous on $(0,1)$?
0
votes
0answers
31 views

Uniform continuity of $\cos x \cos \frac{\pi}{x}$ and $\sin x \cos \frac{\pi}{x}$ in interval $]0,1[$

I have to find the uniform continuity of $f(x)=\cos x \cos \frac{\pi}{x}$ and $g(x)=\sin x \cos \frac{\pi}{x}$ ,where $x \in ]0,1[$ My approach:- For $f(x)$- $$f(x)-f(y)=\sin ...
0
votes
2answers
104 views

Uniform continuity of $x^3+ \sin x$ [closed]

I would like some help in order to analyze the uniform continuity of $f(x)= x^3+\sin x$ over $\Bbb R$. Thank you very much.
4
votes
3answers
82 views

Is $\sin^2x$ uniformly continuous on$x\in [0,\infty]$

I have the question that is $sin^2x$ uniformly continuous on $x \in [0,\infty]$ ? My approach: Let $\left|x-y\right|<\delta$ we have:- $$\left|sin^2x-sin^2y\right|=\left|(\sin x+\sin y)(sin x-sin ...
0
votes
0answers
16 views

Lipschitz condition and differentiability

I have a question related to Lipschitz condition and differentiability. Similar questions have been answered here and here but do not clarify all my doubts. CASE 1: Consider $f:\mathbb{A}\subseteq ...
1
vote
1answer
55 views

Proving an integral is uniformly continuous: $f(x)=\int_{-\infty}^{x} e^{\frac{-t^2}{2}} dt$

Prove that $f(x)=\int_{-\infty}^{x} e^{\frac{-t^2}{2}} dt$ for $x\in\mathbb{R}$ is uniformly continuous Okay so I am unsure how to start this! I can prove a function is uniformly continuous but ...
0
votes
2answers
39 views

Is $ f\left(x\right) =\int^{x}_{0}\sin( e^{t^{2}})\,\mathrm{d}t$ uniformly continuous?

Show that: $$f\left(x\right) =\int^{x}_{0}\sin\left( e^{t^{2}}\right)d\,\mathrm{d}t$$ Is uniformly continuous I have tried to integrate it and bound it by using its limits, but a little unsure ...
1
vote
2answers
37 views

Prove a function is uniformly continuous

Prove the function $f(x)=\sqrt{x^2+1}$ $ (x\in\mathbb{R})$ is uniformly continuous. Now I understand the definition, I am just struggling on what to assign $x$ and $x_0$ Let $\epsilon>0$ ...
0
votes
1answer
35 views

Using the epsilon-delta definition, show that $\tan x$ is not uniformly continuous on $\left[0, \frac{\pi}{2} \right)$.

Fix $\varepsilon$ greater than zero. We need to find $x$ and $y$ such that $|x-y| < \delta$ then $|\tan x - \tan y| > \varepsilon$ for all $\delta$. I am having trouble finding such $x$ and $y$, ...
1
vote
0answers
36 views

Prove that $\phi : C[a,b]\rightarrow \mathbb{R}$ given by $\phi(f)=\int_a^bfdx$ is uniformly continuous

Prove that $\phi : C[a,b]\rightarrow \mathbb{R}$ given by $\phi(f)=\int_a^bfdx$ is uniformly continuous. First, since $f$ is continuous $\phi$ is well defined (integrals exist). Since $f$ is ...
1
vote
4answers
55 views

Assume that $ f: R \to R $ is uniformly continuous. prove that there are constants A,B suchthat $ |f(x)| \le A + B|x| $ for all $ x \in R $.

Assume that $ f: \mathbb R \to \mathbb R $ is uniformly continuous. prove that there are constants $A,B$ such that $ |f(x)| \le A + B|x| $ for all $ x \in \mathbb R $. my concern is just f is ...
6
votes
2answers
590 views

Is a continuous function between two uniformly continuous functions uniformly continuous?

I'm sorry for the long question in the title. Given three functions $\underline{f}(x), f(x), \overline{f}(x)$ that satisfy the following $\underline{f}(x)\leq f(x)\leq \overline{f}(x)$ for all ...
1
vote
1answer
42 views

A function given as an integral is uniformly continuous provided the integrand is uniformly continuous

I need to show that this inequality holds: $| \int_{0}^{1} (h \nabla f(x+sh-y) -h \nabla f(x-y)) ds | \leq |h| \varepsilon(|h|)$ For a function $\varepsilon$ which verifies $\varepsilon (|h|) ...
2
votes
2answers
55 views

Suppose $f : X \to Y$ is a (continuous) bounded map.Does this implies that $f$ is uniformly continuous?

It's well known that if $ f : \bf (X,d) \to \bf (Y,e) $ is a uniformly continuous function then $f$ maps bounded set to bounded set.Does the converse hold ? More Precisely, Suppose $f : X \to Y$ ...
6
votes
1answer
88 views

Let $f: \Bbb R \to \Bbb R$ be a differentiable function such that $\sup_{x \in \Bbb R}|f'(x)| \lt \infty$. Then

$f$ maps a bounded sequence to a bounded sequence. $f$ maps a Cauchy sequence to a Cauchy sequence. $f$ maps a convergent sequence to a convergent sequence. $f$ is uniformly continuous. ...
2
votes
2answers
74 views

Alternative way to show $f(x) = x^2$ is uniformly continuous in a fixed closed interval

We wish to show $f(x) = x^2$ is uniformly continuous in $[a,b]$. This is the way I did it, but it's different from how it's "supposed" to be done. Thus I naturally have some doubts and so here I am. ...
1
vote
2answers
37 views

Show that $\sqrt{x^2 +1 }$ is uniformly continuous. $x \in \mathbb{R}$

Let $x,y \in \mathbb{R} $ such that; If $|x| \leq \delta$ then $|y| = |x| +(|y|-|x|) \leq |x| +||y|-|x|| \leq |x| + |y-x| < 2 \delta$ $$|\sqrt{x^2 +1 } - \sqrt{y^2 +1 }| = \sqrt{y^2 +1 } - ...
1
vote
0answers
24 views

Using Bounded Derivative to Show Uniform Continuity

Suppose $f: (a,b) \to \mathbb{R}$ is differentiable on $(a,b)$ and $|f '(x)|\leq M$ for all $x \in (a,b)$. Prove that $f$ is uniformly continuous on $(a,b)$. My attempt: I started by observing the ...
0
votes
2answers
37 views

Is $f(x)=x\sin(x^{-2})$ uniformly continuous on (0,1)? Is $f(x)=\sin(x^{-2})$?

Is $f(x)=x\sin(x^{-2})$ uniformly continuous on (0,1)? Is $f(x)=\sin(x^{-2})$? Potentially useful equations: For all $\epsilon\gt 0$ $$|f(x)-f(y)|\le\delta$$ $$|x-y|\le\epsilon$$ If the function is ...
1
vote
1answer
41 views

Uniform boundedness implies equicontinuity on compact domains?

Suppose $F:\mathbb R^2 \to \mathbb R$ is continuous. Assume $f_n$ is a uniformly bounded sequence of real-valued functions on $[0,1]$ such that for each $n, f_n'(x) = F(x,f_n(x)),x\in [0,1].$ Is ...