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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On the construction os a space-filling curve by interpolation fractals functions

I'am trying to construct a space-filling curve in the unit square $I^{2}:=[0,1]\times [01,]$ following the results shown in the Barnsley's book "Fractals Everywhere". Thus, let $\Delta:=\big\{ ...
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0answers
10 views

Checking of uniformly continuity of the following functions

Which of the following 4 functions are uniformly continuous? and which are not? I want to know the process/explanation of the solutions.
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1answer
42 views

If $f:[a,b]\rightarrow R$ is a uniformly continuous function then its absolutely continuous?

If $f:[a,b]\rightarrow R$ is a uniformly continuous function then is it true that $f$ is always absolutely continuous?
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1answer
31 views

What is an example of a uniformly continuous function but not absolutely continuous

Is there a function that is uniformly continuous function but not absolutely continuous. My answer is $f(x)=x^{2}, \forall x\in R$ Is this right? Are there any other?
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1answer
46 views

Show continuity or uniform continuity of $\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | )$

$\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | ); \: \: \: \: \: \: \phi(u):=\int_0^1 u^2(t) dt $ Is this function continuous or even uniformly continuous? (I know that the ...
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2answers
24 views

Show that $f$ is uniformly continuous $\iff$ $\bar d(f(x_n),f(y_n)) \to 0$ for all sequences $\{x_n\}, \{y_n\}$ in $X$ with $d(x_n, y_n) \to 0$.

Suppose $(X, d), (Y,\bar d)$ are metric spaces, $f:X \longrightarrow Y$. Show that $f$ is uniformly continuous $\iff$ $\bar d(f(x_n),f(y_n)) \to 0$ for all sequences $\{x_n\}, \{y_n\}$ in $X$ with ...
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43 views

Condition for term-by-term differentiation of a non-convergent series

In a problem I have seen, a series $\sum_n u_n(x)$ with $$ f_n(x) = \frac {\log(1+n^4x^2)}{2n^2}$$ here $\sum u_n'(x)$ is not uniformly convergent, BUT If $f '(x) = \lim_{n\to\infty} f_n'(x) $ ...
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1answer
46 views

Uniform continuity of $\sin(xy)$

Determine whether $f(x,y) = \sin(xy)$ is uniformly continuous on $x,y\geq 0$. Now aside from the definition, I have some trouble even starting with this exercise. I know $|\sin(t)| \leq 1$ ...
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1answer
19 views

Open and closed sets in a $\infty$-metric space

Denote by $\mathcal{H}$ the set of continuous maps $ h:\mathbb{R}\rightarrow \mathbb{R}$ such that $h(0)=0$. We endow $\mathcal{H}$ with the supremum metric $$ \widehat{d}(f,g)=\sup\{\vert ...
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0answers
17 views

Linear programming is continuous

Consider an arbitrary linear program: $$\max \vec c \cdot \vec x$$ subject to: $$\textbf{A}\cdot \vec x = 0, \quad \vec a \le \vec x \le \vec b$$ Assume that this program is feasible and bounded. ...
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2answers
47 views

Show that $f(x)=\frac{x}{1+|x|}$ is uniformly continuous.

I tried to use the definition and arrived this far: $|f(x)-f(y)|=\left|\frac{x}{1+|x|}-\frac{y}{1+|y|}\right|=\frac{|x-y+x|y|-y|x||}{(1+|x|)(1+|y|)}\leq|x-y+x|y|-y|x||$. Any suggestion for ending ...
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4answers
36 views

prove that a non constant periodic, continuous function has a “smallest period”

Let $\ f:\mathbb{R}\to\mathbb{R} \ $ be a non constant, continuous and periodic function. Prove that $f$ has smallest/minimum period. The definition of period that I work with is: $p$ is a period of ...
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1answer
42 views

Uniformly continuous derivative

Let $f:[a,b)\to\mathbb{R}$ be a differentiable function such that $f$ and $f'$ are uniformly continuous in $(a,b)$. Is it true then that $f'$ is continuous at a? Note: By $f'(a)$ I mean ...
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1answer
40 views

Uniform convergence of integral and derivatives of $f_n(x)=\frac{\sin nx}{n}$

Let $f_n$ on $[-\pi,\pi]$ be: $$f_n(x)=\frac{\sin nx}{n}$$ Let $f$ be the pointwise limit of $f_n$ Denote $F(x)=\int_{-\pi}^x f(y)dy$ and $F_n(x)=\int_{-\pi}^x f_n(y)dy$. On $[-\pi,\pi]$: ...
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3answers
40 views

Prove $f(x)$ is not uniformly continuous on $(0,1)$.

Explain or prove why $f(x)=\ln(x)$ is not uniformly continuous on $(0,1)$. By definition $f$ is uniformly continuous if for all $\varepsilon >0$, there exists a $\delta > 0$ such that when ...
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1answer
23 views

Prove a cauchy sequence in $(X,\rho)$ maps to a cauchy sequence in $(Y,\sigma)$

Let $(X,\rho)$ and $(Y,\sigma)$ be two metric spaces. Assume ${x_n}$ is Cauchy in $X$, and that $f:X \rightarrow Y$ is uniformly continuous. Prove that $f(x_n)$ is Cauchy in $Y$. Take ...
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2answers
49 views

Let $f_n(x)=x-x^n$ for $x \in [0,1]$ Does the sequence ${f_n}$ converge pointwise on the set $[0, 1]$? [duplicate]

Let $f_n(x)=x-x^n$ for $x \in [0,1]$ Does the sequence ${f_n}$ converge pointwise on the set $[0, 1]$? This is what I have done, since $0\le x\le 1$ $ x=0$$$f_n(0)=0-0^n=0$$ $x\to \infty$ ...
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2answers
81 views

What do uniformly continuous functions look like?

When I see a function, I want to be able to quickly determine whether it is uniformly continuous or not. Usually, this kind of skill comes after being exposed to many different examples that either do ...
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1answer
15 views

Suppose $f$ is differentiable and not uniformly continuous show that $|f'|$ is not bounded by $R$

Show for any constant $C$, $|f'(t)|> C$ for some $t$ exits in $R$. I can see why this is true but just cannot write up a good solution.
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0answers
7 views

Normal transitional pdf Wiener process continuity mistake and what other standard pdfs are used

I need you to tell me where I am making a mistake in the following: $$f_{1|1}(x_2,t+\Delta t|x_1,t) = \frac{1}{\sqrt{2\pi\Delta t}}e^-{\frac{(x_2-x_1)^2}{2\Delta t}}$$ If I let $\Delta x = x_2-x_1$, ...
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0answers
30 views

continuity of a piece wise function defined partially on a closed interval

using epsilon delta definition prove that $f(x)=\left \{ \begin{array}{cc} 2 & : x \in[0,1]\\ 1 & : x=-1 \end{array}\right.$ is continuous on $E= [0,1] \cup \{-1\}$. Here is my attempt. I ...
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1answer
43 views

Determine continuity and differentiability of the real function $f(x)=\sum\limits_{n\geq1}\frac{1}{n^x}$

I've been asked to analyze the domain, continuity and differentiability of the function $$f(x)=\sum\limits_{n\geq1}\frac{1}{n^x}$$ I've already shown that this function is defined for every real ...
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1answer
50 views

Uniform continuity with integral being finite

Let $f$ be a real valued uniformly continuous function on $\mathbb{R}$ that is lebesgue integrable. Show that $\lim_{|x|\rightarrow \infty}f(x)=0$. Suppose that ...
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1answer
22 views

Polynomials and Lipschitz function

Let $f(x) = x^4 + 11x^2 + 9x -5$ and let $M > 0$. Show that f is a Lipschitz function on the interval $[-M, M]$ I honestly cannot figure out how to start this proof. Nothing similiar is in the ...
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0answers
9 views

Nonuniform Continuity Criteria with cos(1/x)

Use the nonuniform continuity criteron to show that $f(x) = cos\left(\frac{1}{x}\right)$ is not uniformly continuous on $(0, \infty)$ My proof: Let $(x_n)$ be defined by $x_n = \frac{1}{2n\pi}$ ...
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2answers
15 views

Uniformly Continuous functions and paths

Consider a function $ \gamma(t) = (f_1(t), f_2(t), ... , f_k(t))$ for $t \in [a,b]$. So, $f_i$ are real-valued functions on $[a,b]$. Then $\gamma$ is continuous, i.e. a path, if and only if each ...
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40 views

Prove $x^n$ is Uniformly Continuous on a Bounded Subset of $\mathbb{R}$

Hello I want to prove that $f(x)=x^n$ is uniformly continuous on any bounded subset of $\mathbb{R}$. I'm wondering if my proof is correct, and I'm primarily wondering if my choice of $\delta$ is ...
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1answer
27 views

(Uniform) continuity of push-forward operator

I am wondering about extension of the the answer given here. Namely, suppose $U$, $V$ are Polish spaces and $F:U→V$ is uniformly continuous. Does this mean that the push-forward operator $F_*: ...
2
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2answers
92 views

Is the function $f:\mathbb C \to \mathbb C$ defined as $f(z)=\sqrt z$ uniformly continuous ?

Is the function $f:\mathbb C \to \mathbb C$ defined as $f(z)=\sqrt z$ uniformly continuous ? Is it uniformly continuous on some non-compact restriction of the complex plane (other than $(0,\infty)$ ) ...
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0answers
54 views

Give an example of the following or explain why it's impossible (Real Analysis):

A sequence $(f_n)$ of differentiable functions such that both $(f_n)$ and $(f'_n)$ converge uniformly but $f = \lim f_n$ is not differentiable at some point. Thoughts so far: We are taking the ...
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1answer
34 views

Least involved proof for continuos functions => uniform continuos functions on [a,b]

I have been looking at this proof in my textbook and seem to always get lost in its logic, its roughly 3 pages long. The proof is: If f is continuous on a closed interval [a,b], then f is uniformly ...
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1answer
39 views

Bolzano-Weierstrass and Uniform Continuity

I don't really know where to begin here. Any help is appreciated, thank you. "Let K be a compact set in $\mathbb R^n$. You may assume the Bolzano-Weierstrass theorem: every sequence in K has a ...
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2answers
28 views

Uniformly continuous independent of metrics?

Let $(X,d)$ and $(Y,e)$ be metric spaces. A map $f:X\to Y$ is uniformly continuous if for each $\epsilon>0$ there exists $\delta >0$ such that whenever $d(x,y)<\delta$ we have ...
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0answers
22 views

Is it okay if I considered $f$ as uniformly continuous and bounded on specific interval if $f$ is continuous?

Problem Suppose $f$ is continuous and $\phi$ is of bounded variation on $[a, b]$. Show that the function $\psi(x)=\int_a^xfd\phi$ is of bounded variation on $[a, b]$. If $g$ is continous on $[a, ...
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2answers
45 views

If $f_n\to f$ uniformly and $f_n$ are differentiable, is $f$ differentiable?

Suppose $f_n$ converges uniformly to $f$ and $f_n$ are differentiable. Is it true that f will be differentiable? My initial guess is no because $f_n= \frac{\sin(nx)}{\sqrt n}.$ Is this right? And ...
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2answers
38 views

Prove that if $f$ and $g$ are uniformly continuous on A and are both bounded on A, then $fg$ is uniformly continuous on A.

Let $f$ and $g$ be uniformly continuous on A. Then given $\epsilon >0$ there exists a $\delta_{1} > 0$ such that if $|x-y| < \delta_{1}, \forall x,y \in A$, then $|f(x)-f(y)| < ...
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2answers
66 views

How to prove that continuous function do not necessarily preserve cauchy sequences

I am trying to construct a proof that continuous function do not preserve Cauchy sequences Every proof I can find is disprove by counter example, which is great but these counter examples cannot be ...
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73 views

Problem about uniform continunity on $[0,\infty)$

The question is the following: $f(x)$ is uniformly continuous on $[0, \infty)$ and for any $x > 0$, $\lim\limits_{n\to \infty}f(x+n) = 0$, where $n \in \mathbb{Z}_{>0}$. Prove that ...
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42 views

Proof that if $f_n \rightarrow f$ uniformly on $D$, and each $f_n$ is uniformly continuous on $D$, then $f$ is uniformly continuous on $D$.

I came across the following statement - that if $f_n \rightarrow f$ uniformly on $D$, and each $f_n$ is uniformly continuous on $D$, then $f$ is uniformly continuous on $D$. How could I prove this ...
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1answer
63 views

$\forall \epsilon >0 \exists$ a piecewise linear function $g$% such that for a continous function $f$

How would I show the following Edit A function is called piecewise linear if it is (1)Continuous (2)Its graph consists of finitely many linear segments Prove that a continuous function on an ...
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2answers
44 views

continuous and uniformly continuous

I remember proving $x^3$ is not uniformly continuous on $\mathbb{R}$. Then I read the proof of a theorem: Suppose $D$ is compact. Function $f: D \rightarrow \mathbb{R}$ is continuous on $D$ if and ...
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1answer
21 views

Uniform continuity piece wise lienar

A function is called piecewise linear if it is (1)Continuous (2)Its graph consists of finitely many linear segments Prove that a continuous function on an interval [a,b] is the uniform limit of a ...
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2answers
54 views

Is there an uniformly continuous $[0,\infty) \rightarrow \mathbb R$ surjection?

I think there isn't but I can't write a proof. I tried assuming that such function is (as it must be) continuous, and showing that it can't be uniformly cont., but I'm not sure what to do.
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1answer
32 views

Let $f: X \subset \mathbb{R}^n\to \mathbb{R}^m$. Then, $f$ is uniformly continuous if, and only if, for every sequence…

Let $f: X \subset \mathbb{R}^n\to $. Then, $f$ is uniformly continuous if, and only if, for every sequence, $(x_n)_{n\in\mathbb{N}}$, $(y_n)_{n\in\mathbb{N}}$ such that $d(x_n,y_n) \to 0$, then ...
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1answer
52 views

Asymptotics of uniformly continuous functions over $\mathbb R$

For $\alpha >1$, $x^\alpha$ is not uniformly continous over $\mathbb R$ , essentially because $\forall \delta>0, \lim_n (n+\delta)^{\alpha}-n^{\alpha}=\infty$ I don't have a clear intuition ...
0
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1answer
33 views

uniformly continuous functions on infinite open interval $(0, \infty)$

Is $\sin x/x$ uniformly continuous on $(0, \infty)$? Can any one help me proving this? $d/dx(\sin x/x)$ is not bounded on $(0, \infty)$. How to proceed then?
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2answers
32 views

What is the difference between $\implies$ and $\land$ in definition of continuity

Can someone please explain to me what would happen if instead: Given $D \subset \mathbb{R}$ $\forall \epsilon > 0, \exists \delta$ such that $\forall x, x_o \in D, |x-x_o| < \delta \implies ...
0
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1answer
47 views

Showing $\tan x$ is uniformly continuous in $[0,\pi/4]$

Since any function which is continuous on a closed and bounded interval is uniformly continuous, in that way the given function is uniformly continuous. And in the $\epsilon, \delta$ method I tried ...
0
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1answer
46 views

Mean-Value Theorem Hypothesis

I know the mean value theorem hypotheses are: If $f: [a,b] \mapsto R$ is continuous on $[a,b]$ and differentiable on $(a,b)$. But can I still invoke MVT under the hypothesis that: $f: (a,b) \mapsto ...
0
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1answer
44 views

Explain why the following conjecture for $f(x)=[x]+(x-[x])^{[x]}$ is not correct?

Explain why the following conjectures for $f(x)=[x]+(x-[x])^{[x]}$ are not correct ($x\in(0,\infty)$) Conjecture a: $\lim_{x\rightarrow\infty}(f(x)-[x])=0.$ Conjecture b: $f$ is uniformly ...