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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45 views

If $f$ is continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ and if $lim_{x\to\infty}f'(x)=0$ Then $f$ uniformly continuous on $[0,\infty)$

I got this problem: Let $f$ be a continuous function on $[0,\infty)$ and differentiable function on $(0,\infty)$ such that $\lim_{x\to\infty}f'(x)=0$. (1) Prove that for each $0<\epsilon$ there ...
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2answers
83 views

Prove $f$ is uniformly continuous iff $ \lim_{x\to \infty}f(x)=0$

Let $f:[0,\infty)\to (0,\infty)$ be a continuous function and $\displaystyle\int^{\infty}_{0}f<\infty$. Show that $f$ is uniform continuous iff $\displaystyle\lim_{x\to \infty}f(x)=0$ So I ...
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1answer
35 views

Convergence of a sum of sines

If $ s_N(x) := \sum_{n = 1}^N c_n \sin(n x) $ converges uniformly on $[0, \pi]$ as $N \to \infty$ then $c_n = o(n^{-1})$. a) Is $c_n = o(n^{-1})$ sufficient for uniform convergence? b) Is $\sum_n n ...
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1answer
60 views

Modulus of continuity properties and uniform continuity.

Let $f:[a,b]\rightarrow \mathbb{R}$ bounded and $\omega(f,r)=\sup\{|f(x)-f(y)| \colon x,y \in [a,b], \ |x-y|<r\}$ (called modulus of continuity of $f$ EDIT note: the original question is in ...
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2answers
28 views

$f$ is uniformly continuous $\Leftrightarrow$ $\mathrm{Re}f$ and $\mathrm{Im}f$ are uniformly continuous.

Let $f$ be a complex-valued function. In many complex analysis textbook, I have just found that $f$ is continuous $\Leftrightarrow$ $\mathrm{Re}f$ and $\mathrm{Im}f$ are continuous. I wonder that ...
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1answer
36 views

Proving that the function $f(x)=e^\frac{-1}{x}$ is uniformly continuous in $(0,\infty)$

I got this problem: Prove that the function $f(x)=e^\frac{-1}{x}$ is uniformly continuous in $(0,\infty)$ by the definition of uniform continuity. I managed to prove it for $1\leq\epsilon$ as ...
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1answer
37 views

No direct proofs of “if $ f: (X, d_X) \to (Y, d_Y)$ is continuous and $X$ is compact then $f$ is uniformly continuous.”

I am studying the theorem "if $f:(X,d_X)\to (Y,d_Y)$ is continuous and $X$ is compact, then $f$ is uniformly continuous." I am not looking for a proof, but I have an argument against any attempt at a ...
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0answers
25 views

Uniformly continuous function defined on open interval has limits defined at the endpoints.

How do you prove the statement below? Let $f:(a,b)\to\mathbb R$ be a function that is continuous on the bounded, open interval $(a,b)$. Then the two limits $$\lim_{x\to ...
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1answer
44 views

Check uniform continuity of functions

Pick out the uniformly continuous functions: $$ \begin{array}{lll} (a) \quad& f(x) = \cos x \cos \pi x & x \in(0, 1)\\ (b) & f(x) = \sin x \cos \pi x ...
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0answers
21 views

how do we prove uniform continuity for some function

How do we prove if the following function $f(x)=\frac{1}{x}$ $;x>0$ is uniformly continous in $x>\frac{1}{2}$ and not uniformly continous everywhere and $f(x)=x^{n}$ is uniformly continous ...
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2answers
30 views

Prove $f$ isn't uniformly continuous

I already proved (followed by an hint) that $f(y)-f(x) > x(y-x)$ for all $y>x>0$. I need to prove $f$ isn't uniformly continuous on $(0, \infty)$. What I did: Lets assume by contradiction ...
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1answer
47 views

Is there a uniformly continuous function such that $a_{n+1} = f(a_n)$?

Let $a_{n+1} = a_n - a_n^2$ and $a_1 = \frac{2}{3}$. I already proved that $a_n \to 0$ Now I was asked, is there a uniformly continuous function such that $a_{n+1} = f(a_n)$? All I can think of is ...
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2answers
67 views

$\lim_{x \to \infty} f(x)=1 $ $\implies$ $f(x) \sin x$ is uniformly continuous on $\mathbb R$?

Let $f:\mathbb R \to \mathbb R$ be a continuos function such that $\lim_{x \to \infty} f(x)=1 $ , then is it true that $f(x) \sin x$ is uniformly continuous on $\mathbb R$ ?
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1answer
25 views

continuous on $[0,\infty)$ and uniformly continuous on $[a, \infty )$ for some $a\in \mathbb R^+$ , to show uniform continuity on $[0, \infty)$

Let $f:[0, \infty) \to \mathbb R$ be a continuous function which is uniformly continuous on $[a, \infty )$ for some $a\in \mathbb R^+$ , then how to show that $f:[0, \infty) \to \mathbb R$ is ...
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3answers
42 views

A uniform continuity problem

Let $A$ be a set of real numbers and $f:A \to \mathbb R$ be a function such that for every $\epsilon >0$ , there exist a uniformly continuous function $g_\epsilon :A \to \mathbb R$ such that ...
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1answer
66 views

Alternative Uniform-Continuity theorem proof by Luroth

Can please someone elaborately give the proof of Uniform-Continuity theorem ( every continuous function on a closed bounded real interval is uniformly continuous) by Luroth ? thanks in advance
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2answers
58 views

Choosing the right $\delta$ for uniformly continuous function

I'm reading a proof for the claim $g(x)$ is uniformly continuous. It comes down to: $\forall x,y>B:\left|g(x)-g(y)\right|\le \left|x-y\right| + \frac{\varepsilon}{2}$ The auther claims $\delta = ...
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2answers
39 views

Uniform continuity of a subset of $R$

Let $A \subset R$ and $f:A\to R$ be given by $f(x) = x^2$. Then $f$ is uniformly continuous if $A$ is bounded subset of $R$. $A$ is dense subset of $R$. $A$ is unbounded and connected subset of ...
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2answers
26 views

Showing a function is uniformly continuous by using the derivative

Let $f(x) = \sqrt x \cos\frac{1}{x}$. Show $f$ is uniformly continuous on $(0, \infty)$. Now as I recall, we've learned in class that if $f'(x)$ is bounded therefore $f$ is uniformly continuous. ...
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0answers
28 views

Uniformly continuous function which is integrable but does not have a limit [duplicate]

Is there an example of a function $f:[0,+\infty)\to \mathbb{R}$ which is uniformly continuous and $\int_0^{+\infty}|f(x)|dx<+\infty$, but $\lim_{x\to+\infty}f(x)\neq0$ (since it is integrable this ...
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0answers
27 views

Proving $\cos$ is Lipschitz continuous with $L=\frac{\sqrt3}2$ on $[-\frac12,1]$, using $\frac{\sqrt3}2=\cos\frac\pi6=\sin\frac\pi3$

I'm working my way through some analysis exercises to gain a better understanding and I stumbled upon an exercise where I could really use a hint. The task is to show that the inequality $|\cos ...
4
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1answer
69 views

Uniform Continuity, Lebesgue Integrability, and Boundedness

I've been working on this problem whilst studying for a comprehensive exam, and I've come up with a solution that I don't like. I'd appreciate some critiques. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ ...
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1answer
29 views

Continuity of a map to a Frechet space

Let $(A,\| \cdot \|)$ be a normed space and $B$ be a Frechet space equipped with a family $\{ p_k \}_{k \in \mathbb{N}}$ of seminorms. Let $\phi: A \to B$ be a linear transformation satisfying the ...
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1answer
36 views

Uniform continuity and limit

This is related to my other question. Consider the function $a(s)=\dfrac{1}{1+s^2}$. Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $t\mapsto a(t)f(t)$ is bounded uniformly continuous. How ...
2
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3answers
91 views

How to prove, that $e^x$ is uniformly continuous if $x$ is negative?

How can one show with only elementary mathematics, that $e^x$ is a uniformly continuous function on $(-\infty;0]$ I started with $\mid e^x-e^y \mid$, knowing that I assume , that $\mid x-y ...
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1answer
53 views

How to show that a complex-valued function is uniformly continuous?

should a function be uniformly continuous in both arguments if it should be uniformly continuous as a complex-valued function. For example how can I proove that ...
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1answer
72 views

Uniform continuity and translation invariance

Consider the function $a(s)=\dfrac{1}{1+s^2}$ and the space $X=\{f:\mathbb{R}\to \mathbb{R}$ such that $t\mapsto a(t)f(t)$ is bounded uniformly continuous$ \}$. I want to show that $X$ is ...
0
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1answer
77 views

Uniform Continuity $\implies$ Continuity

In metric spaces it is a well known fact that uniformly continuous functions are indeed continuous at any point. What about uniform spaces? How can I prove this? (with the definition of topology in ...
4
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2answers
248 views

Proof of uniform continuity of a function

Show that the function $f(x) = \cfrac{x^2 + 5x - 7}{(x^2 - 9x + 8)(x-2)}$ is uniformly continuous on the interval $(3,5)$ (not with epsilon and delta) How do I do this question? I am sitting an ...
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1answer
48 views

Which function is not uniformly continuous? [closed]

Which of the following functions is not uniformly continuous? $$A.\ \ \ \frac{1}{x}, \ \ \ x \in [1, +\infty)$$ $$B. \ \ \ \ \ \ \ \frac{1}{x}, \ \ \ x \in (1,2)$$ $$C. \ \ \ \ \ \ \ \ \frac{1}{x}, ...
2
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3answers
76 views

$x^{1+\epsilon}$ is not uniformly continuous on $[0,\infty)$

There are two questions. First: is the proof underneath correct? Let $\epsilon>0$ and let $f(x)=x^{1+\epsilon}$. I aim to show that $f$ is not uniformly continuous on $[0,\infty)$. We will show ...
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1answer
67 views

$f_a(x) = e^{ax}$ is uniformly continuous over $[0, \infty)$?

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ defined by $f_a(x) = e^{ax}$. a) Prove that $f(x) = e^x$ is not uniformly continuous. b) Determine for wich values of $a$ the function $f_a(x)$ is ...
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3answers
63 views

Show $f$ is uniformly continuous

Let $f$ continuous function on $[0,\infty)$. Lets assume there are $a,b$ such that: $\lim_{x\rightarrow \infty} f(x)-(ax+b) = 0$. Prove $f$ is uniformly continuous on $[0,\infty)$. Well, At ...
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1answer
49 views

uniform continuity on $(a, b]$ implies limit at $a^+$ exists and finite

Let a uniformly continuous function $f$ on $(a, b]$. Prove that $\lim_{x\rightarrow a^+} f(x)$ exists and finite. What I did so far: from the definition of uniform continuity: ...
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1answer
43 views

Non uniform continuity of the function $\sin\left( \frac{1}{2+\sin(t)+\sin(t\sqrt2)}\right)$

How can we show that the function $$f(t)=\sin\left( \frac{1}{2+\sin(t)+\sin(t\sqrt2)}\right)$$ is not uniformly continuous on $\mathbb{R}$. This question gives a sequence $(t_n)_n$ such that ...
2
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3answers
75 views

Let $f$ be a differentiable function and for all $x$ $f'(x)>x$, prove $f$ isn't uniformly continuous

Suppose $f:(0,\infty)\to \mathbb R$ is differentiable and $f'(x)>x$. Prove that $f$ isn't uniformly continuous in $(0,\infty)$. Hint, prove first that for all $y>x>0$ we have ...
3
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3answers
71 views

Is there a uniformly continuous function $f:\mathbb R \to \mathbb R$ such that $a_{n+1}=f(a_n)$?

Let $a_1=\frac 2 3 , \ a_{n+1}=a_n-a^2_n$ for $n\ge 1$, $a_n$ is monotonically decreasing and bounded: $0\le a_{n+1} \le 1$ and $\displaystyle\lim_{n\to\infty} a_n=0$. Is there a uniformly ...
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0answers
35 views

Uniform Continuous function on $(0,\infty)$ with limit

Let $f$ be uniformly continuous on $(0,\infty)$, and for each $h>0$, $$\lim_{n\to\infty}f(nh)$$ exists. Prove that $$\lim_{x\to\infty}f(x)$$$ exists. My proof is as follows: By uniform continuity ...
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3answers
58 views

Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
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1answer
13 views

A question about Integrability and Uniform Continuity

I got this questions: Prove or disprove by a counterexample the following statements: Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is integrable on every closed interval and let $F(x)=\int_0^x ...
2
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1answer
36 views

uniform continuity of the function $t\mapsto\langle x^*,f(t)\rangle$

Let $X$ be a Banach space. $f:\mathbb{R}\to X$ a function. If we have $t\mapsto\langle x^*,f(t)\rangle$ uniformly continuous on $\mathbb{R}$ for each $x^*\in D$ where $D$ is a dense subset of $X^*$ ...
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2answers
66 views

$f :\mathbb R\to \mathbb R$ is a continuous function of period $1$ , then $f$ is uniformly continuous on $\mathbb R$

Let $f :\mathbb R\to \mathbb R$ be a continuous function such that $f(x+1)=f(x) , \forall x\in \mathbb R$ i.e. $f$ is of period $1$ , then how to prove that $f$ is uniformly continuous on $\mathbb R$ ...
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1answer
56 views

Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous

Working on the following problem from Munkres: Let $(X, d_{X})$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f:A \to Y$ is uniformly continuous, then ...
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0answers
27 views

Show that $f: K_1(0) \rightarrow \mathbb{R}^3$ is Lipschitz.

Firstly, the Assignment: Let $V = (\mathbb{R}^3 ,\|\cdot\|_{\infty})$ where $\|\cdot\|_{\infty}$ denotes the maximum norm and consider the function: $$f: K_1(0) \rightarrow ...
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0answers
36 views

Why is this function uniformly continuous?

Why is this function uniformly continuous? $f:\mathbb{R}^2 \rightarrow\mathbb{R}$, $f(x,y)=e^{-(x^2+y^2)}$. I tried using the mean value theorem for several variables, but I am not able to complete ...
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2answers
46 views

Is the set of translations of a function compact?

Let $X=BUC(\mathbb{R})$ be the Banach space of real bounded uniformly continuous functions on $\mathbb{R}$ equipped with the supremum norm. Let $f\in X$, then the subset $$\{f_a:t\mapsto f(t+a), \ \ ...
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1answer
105 views

How to prove uniform continuity problem!

A) $f(x)=x^3$ , give an example of an interval where $f$ is uniformly continuous and another where it is not. explain your choose of examples B) decide if $f(x)= \dfrac{1}{\sin x} - \dfrac{1}{x}$ is ...
2
votes
1answer
27 views

$\lim_{t\to 0}\sup_{s\in \mathbb{R}}|f(t+s)-f(s)|$ uniformly on $f$

Let $X$ be the space of all bounded uniformly continuous functions $f:\mathbb{R}\to \mathbb{R}$ equiped with the supremum norm $|f|_\infty$. We know that for each $f\in X$ we have $$\sup_{s\in ...
2
votes
1answer
80 views

Is $f(x)=\log(1+x^2)$ uniformly continuous on $(0,\infty)$?

Is $f(x) = \log(1+x^2)$ uniformly continuous on $(0,\infty)$? My work: Looking at the graph and knowing that $\log$ considered a "slow-growing" function, my guess is that $f(x)$ is uniformly ...
5
votes
2answers
434 views

Does there exist an unbounded function that is uniformly continuous?

I know that $1/x$ is unbounded on $(0,5)$ (for example) and that since it is unbounded, it is not uniformly continuous. Does a function have to be bounded to be uniformly continuous? I don't think ...