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 the composition of uniformly continuous functions is uniformly continuous.

Let $X,Y,Z$ be subsets of $\textbf{R}$. Let $f:X\rightarrow Y$ be a function which is uniformly continuous on $X$, and let $g:Y\rightarrow Z$ be a function which is uniformly continuous on $Y$. Show ...
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1answer
35 views

How do we gain a better intuition on the definition of uniform continuity and its advantages compared to usual continuity?

So here I am studying uniform continuity. Besides its definition, it has been proved that uniformly continuous functions map pairs of equivalent sequences to pairs of equivalent sequences, Cauchy ...
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1answer
55 views

Let $f:X\rightarrow\textbf{R}$ be a uniformly continuous function. Suppose that $E$ is a bounded subset of $X$. Then $f(E)$ is also bounded.

Let $X$ be a subset of $\textbf{R}$, and let $f:X\rightarrow\textbf{R}$ be a uniformly continuous function. Suppose that $E$ is a bounded subset of $X$. Then $f(E)$ is also bounded. MY ATTEMPT I ...
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26 views

How do we prove that uniformly continuous functions map Cauchy sequences onto Cauchy sequences?

Let $X$ be a subset of $\textbf{R}$, and let $f:X\rightarrow\textbf{R}$ be a uniformly continuous function. Let $(x_{n})_{n=0}^{\infty}$ be a Cauchy sequence consisting entirely of elements in $X$. ...
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38 views

Relation between uniform continuity and equivalent sequences: how do we prove they are related?

Let $X$ be a subset of $\textbf{R}$, and let $f:X\rightarrow\textbf{R}$ be a function. Then the following two statements are logically equivalent: (a) $f$ is uniformly continuous on $X$ (b) Whenever ...
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1answer
63 views

Prove that the function f(x)=x^2 is not uniformly continuous on F: C[0,1]→C[0,1]. [closed]

So it has to be NOT uniformly continuous. But let $x_1, x_2 \in F$. $0 < x_1 < 1$ and $0 < x_2 < 1$. Let $\delta = \varepsilon/2$ and $|x_1 - x_2| < \delta$. Then we have: $|f(x_1)...
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1answer
102 views

Show that a uniformly continuous function on $E$ has a unique continuous extension to $cl(E)$

Suppose X is a metric space and $f : E ⊂ X → R$ is a uniformly continuous function on a set E. Denote cl(E) to be the closure of E in X. Prove that there is a unique continuous function $g : cl(E) → R$...
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32 views

Uniformly continuous or not?

So I supposed to find out if $$f(x)=\frac{1}{1+\ln^2 x}$$ is uniformly continuous on $I=(0,\infty)$ So I have been thinking a lot. Could I say that $f$ is continuous on $[0,1]$ and therefore uniformly ...
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30 views

Show that a given composite function is continuous

Question Let $f : D \rightarrow \mathbb R$ and $g : E \rightarrow \mathbb R$ be two uniformly continuous functions with $f(D) \subseteq E$. Show that the composite function $g \circ f : D \rightarrow\...
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1answer
47 views

An attempt at proving “continuous function on a closed interval (I) is uniformly continuous”

I am familiar with some of the standard proofs of the statement. However, I was trying to construct a proof that fits best with my natural intuition. To this end, given $\varepsilon >0$, I define ...
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43 views

Show that $\frac{1}{x+2}$ is not uniformly continuous for $(-2,0]$.

I need help proving that $\frac{1}{x+2}$ is not uniformly continuous for $(-2,0]$, using an $ε-δ$ proof. I understand that intuitively this is just the function $\frac{1}{x}$ which is not uniformly ...
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1answer
56 views

A uniformly continuous function can be extended on the boundary

Suppose $X$ a metric space, $Y$ a complete metric space and $f: S \rightarrow Y$ a uniformly continuous function from $S \subseteq X$ to $Y$. Prove that $f$ can be extended to a uniformly continuous ...
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53 views

Does $\lim_{x\to\infty}|f(x+h)-f(x)|=0$ implies $f$ is uniformly continuous?

Question: Let $f\in C([0,\infty))$. And $\forall \ h\in\mathbb{R}$, $$ \lim_{x\to\infty}|f(x+h)-f(x)|=0. $$ Show that $f$ is uniformly continuous on $[0,\infty)$. I have some idea about this ...
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30 views

Prove that linear bounded functional on subspace of C[a,b] that does not attain its norm

Let $L$ be a subspace of space $C[a,b]$ such that $ || f|| = sup_{t\in [a,b]}|f(t)|$. L consists of all $f \in C[a,b]$ such that $ \int_{a}^{\frac{a+b}{2}} f(x)dx=\int_{\frac{a+b}{2}}^{b} f(x)dx $...
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28 views

Prove that $f$ is uniformly continuous on $(-1,3)$ using the definition of UC.

$$f(x)= \left\{ \begin{array}{ll} \frac{2-\sqrt{4-x}}{x} & x\neq 0 \\ \frac{1}{4} & x=0 \end{array} \right.$$ Either $x\neq 0$ and $y\neq0$, $x=0$ and $y\neq 0$, $x\neq 0$ and $y=...
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1answer
39 views

Show that $x \sin x$ is not uniformly continuous

I have seen a lot of posts that have solved this already by taking two sequences $(a_n), (b_n)$, then showing $\lim(a_n - b_n) = 0$ but $|f(a_n) - f(b_n)| > \epsilon$, but I would like to show ...
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3answers
88 views

Prove that if $\frac{f'(x)}{x}$ is bounded, then $g := \frac{f(x)}{x}$ is uniformly continuous

So, i've been trying to solve this problem. If $f$ is a function from $[1,\infty)$ to $R$, such that $ \frac{f'(x)}{x}$ is bounded, then $ g(x) := \frac{f(x)}{x}$ is uniformly continuous in $[1,\...
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1answer
25 views

Show whether the following functions are Lipschitz or not. [closed]

Let $f_1 : \mathbb R \to\mathbb R $ be defined as $f_1 = |x|$. Show that $f_1$ is Lipschitz on $\mathbb R $. Let $f_2 : [a,b] \to\mathbb R $ be defined as $f_2 =x^2 $. Show that $f_2$ is Lipschitz on ...
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1answer
34 views

uniform continuous in $F:(C[0,1],d_1)→(c_0,d_2)$

Let $f:[0,1]\to\Bbb R$ be a countinous fuction, $$d_1(f,g)=\sup|f(x)-g(x)|$$ and $$d_2(x,y)=\sup|x_n-y_n|$$ and $$c_0 = \{ (x_n) \subseteq \mathbb{R} :\lim \;x_n = 0)\}$$ Now for $f\in C[0,1]$, we ...
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Does the limit of this integrand (product of monotonic functions) exist, given the integral is bounded?

For $p$, $f$ sufficiently differentiable functions: \begin{equation} \int_{-\infty}^{\infty} f^2 p = 1 \qquad \text{(i.e. $f \in L^2_p$)} \end{equation} I want to conclude $\lim_{x\to\pm\infty} f^2 p ...
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28 views

Quick doubt on this exercise of Uniform Continuity

Problem : Let $X,Y,Z \subset \mathbb R$ and assume that $f:X\to Y$ is uniformly continuous on $X$ and $g:Y\to Z$ is uniformly continuous on $Y$. Then $(g \circ f):X \to Z$ is uniformly continuous ...
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1answer
30 views

For $A \subset \mathbb{R}$, the function $f: A \to \mathbb R$, $f(x)=x^2$ is a uniform continuous function if?

For $A \subset \mathbb{R}$, the function $f: A \to \mathbb R$, $f(x)=x^2$ is a uniform continuous function if (A) A is bounded (B) A is unbounded (C) A is compact (D) A is the set $\mathbb Z$ of ...
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1answer
30 views

Which of the following is(are) false?

For $ E \subset \mathbb{R}$, consider the following statements: $P$: Every continuous function $f: E \to \mathbb{R}$ is uniformly continuous. $Q$: $E$ is compact. $R$: Every continuous function $f: ...
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1answer
59 views

Is Shadowing Lemma specific to hyperbolic dynamical systems?

In a hyperbolic dynamical system, the Shadowing lemma states that every epsilon-pseudo-orbit is uniquely delta-shadowed by some orbit. see: https://en.wikipedia.org/wiki/Shadowing_lemma It is not ...
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45 views

Finding a necessarily and sufficient condition for a uniform continuity of a series

Consider the function $f:\Bbb R\to \Bbb R$ given by $f(x)=\max\{0,1-|x|\}$, and let $(s_n)$ be a sequence of real numbers. I am asked to show to find a necessarily and sufficient condition for the ...
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3answers
37 views

Prove a certain cubic function is uniformly continuous on the unit interval

Let $f:[0, 1] \to \mathbb{R}$, $f(x) = -x^3+2x$. Prove using the definition of uniform continuity that the function is uniformly continuous. By definition $\forall \varepsilon >0$ $\exists \delta&...
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2answers
58 views

If $f$ is a uniformly continuous function from $(0,\infty) \rightarrow (0,\infty)$. Then is $\lim_ {x \rightarrow \infty} f(x+1/x)/f(x) = 1$?

If $f$ is a uniformly continuous function from $(0,\infty) \rightarrow (0,\infty)$. Then is $\lim_ {x \rightarrow \infty} f(x+1/x)/f(x) = 1$ ? What I have tried so far is Since $f$ is uniformly ...
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38 views

Errors in incorrect proof that a function is lipschitz continuous iff it is uniformly continuous.

Proposition: a function $f: \mathbb{R} \to \mathbb{R}$ is uniformly continuous if and only if it is Lipschitz continuous. Proof: Suppose $f$ is uniformly continuous. Then, by the definition of ...
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1answer
29 views

Proving a recursive sequence is convergent to a fixed point of a function

Problem: Let $f:[a,b]\to[a,b]$ be a function such that $|f(x)-f(y)|\leq|x-y|$ for every $x,y\in[a,b]$. Choose $x_1\in[a,b]$ and define $x_{n+1}=\frac{x_n+f(x_n)}{2}$. Prove that the sequence $\{x_n\}$ ...
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26 views

Is there a way to tell if a function is uniformly continuous by taking a look at the equation of the function?

I was introduced to the concept of uniform continuity using the definition of epsilon-delta earlier last week. While I get the concept of what it should mean, I would like to ask if anyone has an ...
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18 views

Real analysis. Continuity of a product [duplicate]

Let $X$ $\subset$ $\mathbb{R}$ and $f,g:X\rightarrow\mathbb{R}$ two continuous bounded functions. Is the product $fg$ a uniformly continuous function? Prove or give a counterexample. I would say it ...
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41 views

What can be wrong in this ~*proof*~ that $f(x)=x\sin x$ is uniformly continuous?

So I thought I had solved an assignment, but there are good proofs of the opposite on the Internet. I am confused entirely. Suppose $f:(0, \infty) \to \mathbb{R}, f(x) = x\sin x$. Let $x_n, y_n$ be a ...
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12 views

Continuity and differentiability of $\ \frac{\arctan(n^{\frac{1}{4}}x^2)}{n^{\frac{3}{2}}} $

I have to prove uniform continuity abd differentiability for $$\ f_n(x)= \frac{\arctan(n^{\frac{1}{4}}x^2)}{n^{\frac{3}{2}}} $$ So far I have estimated in order to find uniform continuity $$\ \...
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1answer
31 views

Is $X$ compact if every continuous real valued function on $X$ is uniformly continuous and $X$ has finitely many isolated points.

There was a problem of metric space which I recently found. It is as follows: Suppose $X$ is a metric space with finitely many isolated points and having the property that every real value continuous ...
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1answer
77 views

If $f : (a,b) \rightarrow \Bbb R$ is uniformly continuous then it is bounded

I have an exercise in my textbook: The first one I try to prove by contrapositive: Suppose $f$ is not bounded. Then $\forall M \in \Bbb R : |f(x)| \gt M \text{ },\forall x \in X$ $\Rightarrow f(x) \...
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Is the map $f \mapsto \nabla f(x)$ continuous over the space of convex functions?

Take $V=C(X,\mathbb R)$ as the space of differentiable functions $X \to \mathbb R$ under uniform convergence. Here $X \subset \mathbb R^d$ is some compact domain. For each x is well known the map $\...
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1answer
43 views

Which of the following statements about f is true?

For the function $f (x)$ on the real line $\mathbb{R}$ defined below, which of the following statements about $f$ is true?Choose all the correct options: $$f (x) :=\sum_{n\ge 1}\frac{\text{sin}(x/n)}...
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2answers
65 views

Is it true that $f(a+x)-f(x)$ is bounded for any $a$ if $f(x)$ is uniformly continuous on $\mathbb{R}$

I can think of a counter-example that when $f(x)=x$, then $f(a+x)-f(x)=a$ is not bounded as $a$ is getter larger and larger. But what if I only require the bound to be independent of $x$? Can I ...
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1answer
26 views

If $f$ is continuous from a compact metric space to real numbers, prove $f^2$ is uniformly continuous

let $(X,d)$ be a COMPACT metric space. and $f$ is continuous function on $X$ that maps $X$ to real numbers, prove $f^2$ (pointwise product) is uniformly continuous. I know by theorem, that $f$ is ...
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3answers
39 views

Which of the following are uniformly continuous over $\mathbb{R}$?

Which of the following are uniformly continuous over $\mathbb{R}$? (i) $f(x)=\int_{0}^{x}g(t)dt$, where $g:\mathbb{R}\to\mathbb{R}$ is a continuously differentiable function. (ii) $f(x)=\int_{0}^{x}g(...
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0answers
14 views

Relation between $C^k(\Omega)$ and $C^{k-1}(\bar{\Omega})$

Let $\Omega$ be a bounded open domain. Let k be a positive integer. $C^k(\Omega)$ is the space of functions that are continuous derivatives up to order $k$ on $\Omega$, and $C^k(\Omega)$ will denote ...
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1answer
36 views

Clarity on uniform continuity of functions

Show that $f(x) = \text{cos}x\text{cos}\frac{\pi}{x}$, $x\in(0,1)$ is not uniformly continuous while $g(x) = \text{sin}x\text{sin}\frac{\pi}{x}$, $x\in(0,1)$ is uniformly continuous on the given ...
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1answer
53 views

Which of the following are uniformly continuous?

Which of the following functions are uniformly continuous over their respective domain of definition? (a) $f(x) = \text{cos}x\text{cos}\frac{\pi}{x}$, $x\in(0,1)$ (b) $g(x) = \text{sin}x\text{sin}\...
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2answers
27 views

If f+g is uniformly continuous on a subset A of R then are f and g uniformly continuous??

I have been going throught Bartle Sherbert book and I found the problem where it was asked to prove that f+g is uniformly continuous and so on. I was thinking about the converse of that statement. Is ...
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1answer
26 views

question about uniformly continuity

If $f$: $D$-->$E$ and $g$: $E$-->$R$ are uniformly continuous, is $g º f$: $D$-->$R$ also uniformly continuous? I think it is also uniformly continuous but I don't know how to write reasoning. can I ...
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1answer
28 views

What condition do I need else to show a uniformly continuous is pointwise bounded and uniformly bounded?

Let a sequence of function f_n uniformly continuous is given. I am wondering what else I need to let it be pointwise bounded and what for uniformly bounded. I am confused with some knowledge about it. ...
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1answer
29 views

Show that $g$ is well-defined and that $g$ is monotone continuous function.

Let $f : [0, 1] \to\mathbb R $ be a continuous function. Define $g(0) = f(0)$ and $g(x) = \max\{f(y) \mid 0 ≤ y ≤ x \}$ for $0 < x ≤ 1.$ Show that $g$ is well-defined and that $g$ is monotone ...
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1answer
30 views

prove f+g uniform continuity

Suppose the functions $f$ and $g$ are uniformly continuous on $D$. prove that $f+g$ is also uniformly continuous continuous on $D$. Show that $f⋅g$ is not necessarily uniformly continuous by a ...
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1answer
37 views

Continuity on a subset vs continuity on a subspace.

Suppose $f:X\to Y$ is a function between two metric spaces such that $f:X\to Y$ is continuous on $A$ i.e. at each point of $A$.But is this equivalent to saying $f|_A:A\to Y$ is continuous between the ...
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1answer
17 views

A question on continuous extension.

Suppose $X$ and $Y$ are metric spaces and $A$ is a dense subset of $X$ and assume $Y$ is complete.Suppose $f:A\to Y$ is a continuous function.Then a natural question arises whether this can be ...

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