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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Continuity on $[a,b]$ implies uniform continuity on $[a,b]$

I don't understand the step underlined in green. I understand that for any $n$ , $|f(x_n)-f(y_n)|\geq \varepsilon$ where $x_n, y_n$ satisfy the conditions given regarding a function being not ...
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Negation of uniform continuity

The definition of uniform continuity is: Given any $\varepsilon>0\ \exists\delta>0\ \forall x\in I \ \forall y\in I\ \left(\text{if }|x-y|<\delta\text{ then }\ ...
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32 views

Proof: Cauchy sequences and uniform continuity

I'm working on a proof and I'm having trouble relating definitions I want to prove that if f is uniformly continuous, then if a sequence $ {a_n} $ is Cauchy, $ {f(a_n)} $ is Cauchy. So if $ f $ is ...
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If $f,g$ are unifromly continuous, then $\alpha f+\beta g$ is uniformly continuous?

If $f,g$ are uniformly continuous, then is $\alpha f+\beta g$ uniformly continuous? So far, I looked at here If $f,g$ are uniformly continuous prove $f+g,fg$ are uniformly continuous, but I didn't ...
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$\lim_{x\to x_0 ;x\in X} f ( x)$ exists if f is a uniform continuous function and $x_0$ is an adherent point

Proposition: Let $X$ be a subset of $R$, let $f:X\to R$ be a uniformly continuous function, and let $x_0$ be an adherent point of $X$. Then $\lim_{x\to x_0 ;x\in X} f ( x)$ exists. Proof Take any ...
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44 views

Prove a Continuous Distribution Function is Uniformly Continuous

Let $F$ be the distribution function for a random variable $X$ and it is given that $F$ is continuous over the entire real line. Prove that $F$ is uniformly continuous over the real line. My ...
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counter example of equicontinuous

Consider the functions on $[0,1]$: $f_n(x)=nx$, when $x$ is between $0$ and $1/n$ $f_n(x)=2-nx$, when $x$ is between $1/n$ and $2/n$ $f_n(x)=0$, otherwise How to see it is not (uniformly) ...
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Relations between $\varepsilon$ and $\delta$ in the $\varepsilon-\delta$-Defintion of Continuity

A function $f : \mathbb R \to \mathbb R$ is continuous at $x_0$ iff for each $\varepsilon > 0$ there exists some $\delta > 0$ such that $$ |x - x_0| < \delta \Rightarrow |f(x) - f(x_0)| < ...
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why $\sin (x \sin x)$ is not uniformly continous for $x>0$ .

I think it should be since for $(0,1]$ we can use continous extension theorem and for $x>1$ it is lipschitz (since it lies below $y=x$ line).What is wrong in the argument? This argument also proves ...
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36 views

uniformly continuous function is bounded

I saw already someone proved this question. for a bounded set $E$, if $f$ is uniformly continuous on $E$ then $f$ is bounded on $E$ proof what I saw is by dividing $E$ into $n$ ...
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Uniform continuity and lipschitz condition

Let $f(x)=\frac{x^2}{x^2+1}$ where $f:\mathbb{R} \to \mathbb{R}$ examine uniform continuity and lipschitz condition. I'm not sure if my solutuion is OK. $\displaystyle \ \ |f(x)-f(y)|= ...
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1answer
32 views

Uniform continuity of two functions

Investigate uniform continuity of the following functions: $$a) \ f(x)=\frac{1}{x} \\ b) \ f(x)=\cos \frac{1}{x}$$ How to deal with such questions, i have little knowledge about that topic thus i ...
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If $f$ is continuous & $\lim_{|x|\to {\infty}}f(x)=0$ then $f$ is uniformly continuous

Let, $f:\mathbb R\to \mathbb R$ be a continuous function such that $\lim_{|x|\to {\infty}}f(x)=0.$ Then prove that $f$ is uniformly continuous. I tried through the formal definition of uniform ...
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Proof that a certain function is uniformly continuous?

Consider a metric space $(M, {\rm d})$ and $y$ fixed in $M$. I want to prove that the function $f$ defined by $f(x)\colon={\rm d}(x,y)$ is uniformly continuous. So I know that if this function ...
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How to show that “Uniformly continuous implies continuous”? [closed]

Can I go from the definition of uniformly continuity to continuity? Please somebody show me how to do that. Thanks.
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60 views

how to prove $f(x)= x^2$ is uniformly continuous on $\mathbb{N}$

Question is $f(x)=x^2$ is uniformly continuous on $\mathbb{N}$ I know $f$ is uniformly continuous on bounded set because $\delta$ can be found easily from ...
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Is $\;\frac{\ln(x+1)}x\;$ uniformly continuous in $\;[0,\infty)\;$?

I have this problem: is the function $$f(x)=\frac{\ln(x+1)}x$$ uniformly continuous in $\;[0,\infty)\;$ ? First, I think I could say that zero is not even in domain of function, but I can show it ...
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28 views

Help proving a theorem on uniform continuity in an open interval

I need help proving that given f:(a,b)→R that is uniformly continuous, it is possible to extend f to f:[a,b]→R that is continuous on the closed interval. thanks in advance!
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73 views

The function $\varphi(u)=\sup_{x\in[0,1]}(f(x)+u\ g(x))$ is Lipschitz continuous

Let $f$ and $g$ two function bounded such that : $$f: [0,1] \to \mathbb{R}$$ $$g: [0,1] \to \mathbb{R}$$ Let : $$\varphi(u)=\sup_{x\in[0,1]}(f(x)+u\ g(x)),\quad u \in \mathbb{R}$$ Show ...
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39 views

A property of uniformly continuous functions

We know that if $f:\mathbb{R}\to\mathbb{R}$ is a function such that for all $x,y\in \mathbb{R}$ $$|f(x)-f(y)|\leq a(|x-y|),$$ where $a:\mathbb{R}\to\mathbb{R}$ is a function independent of $x$ and $y$ ...
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28 views

Prove uniform continuity of function

I was given $f: <1,+\infty>\times<1,+\infty>\rightarrow <0,+\infty> $ defined with $f(x,y)=\ln x+\ln y$ and metric on both spaces is induced by taxicab norm. I need to prove this ...
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uniform continuity in $(0,\infty)$

Prove uniform continuity over $(0,\infty)$ using epsilon-delta of the function $$f(x)=\sqrt{x}\sin(1/x)$$ I know that I start with: For every $\varepsilon>0$ there is a $\delta>0$ such that ...
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In kernel rules, why is a regular kernel necessarily bounded?

Request from a dabbler in measure theory. Here a kernel merely designates a real-valued function. Quoted from "A Probabilistic Theory of Pattern Recognition", Luc Devroye, Laszlo Gyorfi, Gabor ...
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30 views

Uniform continuity of a function $F:(0,1)\times (0,1)\to \mathbb R$

Let, $f,g:(0,1)\times (0,1)\to \mathbb R$ be two continuous functions defined by $f(x,y)=\dfrac{1}{1+x(1-y)}$ & $g(x,y)=\dfrac{1}{1+x(y-1)}$. Then which is correct? (a) $f$ & $g$ both are ...
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62 views

uniform continuity and equivalent sequences

Let $X$ be a subset of $\mathbb{R}$, and let $f : X\to \mathbb{R}$ be a function. Then the following two statements are logically equivalent: (a) $f$ is uniformly continuous on $X$. (b) ...
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63 views

Is $f(x)=x\sin x$ uniformly continuous in the interval $(0,a)$ when $a>0$?

Is $f(x)=x\sin x$ uniformly continuous in the interval $(0,a)$ while $a>0$? I have proven that its not uniformly continuous in the interval $[0,\infty)$ because the function "$x\sin x$" is ...
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Can a function be continuous but not Hölder on a compact set?

Is it possible to construct a function $f: K \to \mathbb{R}$, where $K \subset \mathbb{R}$ is compact, such that $f$ is continuous but not Hölder continuous of any order? It seems like there should ...
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Bounded partial derivatives implies uniform continuity

As is well-known, by mean value theorem, if $f$ is continuously differentiable on a convex domain $U$, and $Df$ is bounded, then $f$ is Lipschitz continuous (and thus uniformly continuous). However, ...
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Uniform continuity of the antiderivative

We know that if $f:\mathbb{R}\to\mathbb{R}$ is a function such that $$\sup_{x\in\mathbb{R}}|f(y)|<\infty,$$ then the function $g(x)=\int_0^xf(y)dy$ is uniformly continuous. I am just wondering ...
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Finding $r$ such that $|f(x,y)| \leq 1/4$ whenever $(x,y) \in [0,1] \times [-r,r]$.

Let $f: \Bbb R^2 \to \Bbb R$ be a continuous function such that $f(x,0) = 0,$ for all $x$. Show that exists $r > 0$ such that $|f(x,y)| \leq 1/4$ whenever $(x,y) \in [0,1] \times [-r,r]$. ...
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Continuous Nowhere Differentiable Function [closed]

Define a function $\,f:\mathbb{R}\rightarrow \mathbb{R}_{+}$ by: $$ f(x)=\left|x-2\,\left \lfloor \frac{x+1}{2}\right \rfloor \right|. $$ Here are some known properties about the function $f$: ...
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Show that $f(x) = x\cos^3(x)$ is not uniformly continuous on $\mathbb{R}$

Show that $f(x) = x\cos^3(x)$ is not uniformly continuous on $\mathbb{R}$. I tried $x_n = \pi/2 + n\pi$ and $y_n = \pi/2 + n\pi + 1/(n\pi)$. Since $\cos^3(x) = \frac14 (\cos(3x)+3\cos x)$, ...
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Uniform convergence of the sequence $f_n(x)=f(x+1/n)$ for uniformly continuous $f$

Let $f$ be a uniformly continuous real-valued function on $(-\infty, +\infty)$, and for each $n\in I$ let $f_n(x)=f\left(x+\frac{1}{n}\right)$. Prove that $\{f_n\}_{1}^{\infty}$ converges ...
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Is it true that a continuous function with compact support is uniformly continuous?

I've been trying to prove the given $f:\mathbb R\rightarrow \mathbb C$ continuous with compact support, $f$ is uniformly continuous. I don't know if it's true or not, but it is highly plausible and ...
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74 views

Extending a uniformly continous function to the closure of its domain

Suppose $X$ is a noraml space and $f:X \rightarrow X$ is continous on X, and also uniformly continous on a subset $A \subseteq X$. In this setting, can one conclude that f is uniformly continous on ...
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Is function $f$ also uniformly continuous?

I've been thinking on the following problem lately: Let $(X,d)$ be a metric space and $f_1,f_2,...,f_n: X \rightarrow \mathbb{R}$ and $f(x) = \max\{f_1(x),f_2(x),...,f_n(x) \}$,$x\in X$ If the ...
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Proving uniform continuity and uniform discontinuity

Could someone please explain to me how to show uniform continuity and not uniformly continuous for the following: $f(x) = \frac{1}{x^2}$ for $A = [1, \infty)$ show uniform continuity $f(x) = ...
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Need to show the following function is uniformly continuous on R

Could you please tell me how I am supposed to show that $f(x) = \dfrac{1}{(1+x^2)}$ is uniformly continuous in $\mathbb{R}$. I did some pre-calculation and found that $|f(x) - f(u)| < \epsilon$ if ...
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A concave positive function on $[1,\infty)$ is uniformly continuous

Let $f$ be a concave positive function on $[1,\infty)$, then $f$ is uniformly continuous on $[1,\infty)$. This was a true or false problem that I couldn't prove to be true, so I'm thinking that maybe ...
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Continuous functions which are Uniformly continuous

Let $f:[a,\infty)\rightarrow \mathbb{R}$ be continuous such that $lim_{x\rightarrow \infty} f(x)$ exists(say $c$). Show that $f$ is uniformly continuous on $[a,\infty)$. My work: Let $\epsilon ...
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Prove $g(x)= x^2 \sin \left(\frac1x\right)$ is not uniformly continuous

Prove $g(x)= \{x^2 \sin \left(\frac1x\right);x\neq 0$ and $g(0)=0$ is not uniformly continuous so I know g is continuous at $0$. However, I think $g$ isn't uniformly continuous based on the graph. ...
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If $f$ is uniformly continuous on $[0, \infty)$ [duplicate]

If $f$ is uniformly continuous on $[0, W)$ for all $W>0$ and $lim_{x\to\infty}f(x)=L$, prove that $f$ is uniformly continuous on $[0, \infty)$. I understand that this means $f$ is uniformly ...
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43 views

Are derivatives of Schwartz functions uniformly continuous?

Let $f\in\mathscr{S}(\mathbb R^n)$ where $\mathscr{S}(\mathbb R^n)$ is the Schwartz space. Is it true that $\partial^\alpha f$ is a uniformly continuous function for all multi-index $\alpha$? Here ...
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Prove $\{g(x_n)\}_{n=1}^\infty$ converges

Let $g : (a, b) → R$ be uniformly continuous on $(a, b)$. Let $\{x_n\}_{n=1}^\infty$ be a sequence in $(a, b)$ converging to $a$. Prove that $\{g(x_n)\}_{n=1}^\infty$ converges. The general idea here ...
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Uniform Continuity and Limits

Let $f\colon (0, 1]\to\mathbb{R}$ be a function which is uniformly continuous. (i) Show that if $\{x_n\}$ is a sequence with $x_n > 0$ and $\lim_{n\to∞} x_n = 0$, then $\{f(x_n)\}$ is convergent. ...
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88 views

If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$

Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints. I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} ...
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26 views

Prove that if f is an element of Lip a and a>1 then f is constant

We have that a function $f: [a,b]\mapsto\Bbb R$ satisfies a Lipschitz condition of order $a > 0$ if there is some positive constant $M$ so that $$|f(x_1)-f(x_2)| \leq M|x_1-x_2|^a \qquad \forall ...
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1answer
41 views

Prove that if f is an element of Lip a, then f is uniformly continuous

I am required to prove that if f is an element of Lip a, then f is uniformly continuous. Where Lip means a Lipschitz function. I also have to prove that if f is an element of Lip a and a>1 then f is ...
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100 views

Show that $g(x)=\sqrt{x}$ is uniformly continuous on $[0, \infty)$

I must show that $g(x)=\sqrt{x}$ is uniformly continuous on $[0, \infty)$. I know we have to show that $\sqrt{a-b}\ge \sqrt{a} - \sqrt{b}$ and $\sqrt{a + b} \le \sqrt{a} + \sqrt{b}$ but I don't know ...
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31 views

Conditions for open interval continuity

Please can someone help in giving me the condition that would make a continuous function on an open interval be uniformly continuous in that same interval. Thanks.