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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18 views

Confusion regarding uniform continuity

I was trying to check the validity of the following: If $f:\mathbb R\rightarrow\mathbb R$ and its derivative $f'$ are unbounded, then $f$ is not uniformly continuous on $\mathbb R$. To me,the ...
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1answer
19 views

Testing if a function is uniformly continuous on an open interval

Let $f(x)=\sqrt[3]{4x}\sin\left(\frac{5}{x^2}\right)$, is this uniformly continuous on $(0,2)$? I started out using the Weierstrass M-Test but quickly realized that this does not apply since I don't ...
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2answers
59 views

Showing $f(x_1,x_2) = \frac{xy}{x^2 + y^2}$ is uniformly continuous on $\{x \in \mathbb{R}^n : \left\|x\right\| \geq r\}$.

Let $f : \mathbb{R}^2 \setminus \{0\} \to \mathbb{R}$ be defined by $$f(x_1, x_2) = \frac{x_1x_2}{x_1^2 + x_2^2}. $$ I want to show that f is uniformly continuous on $\{x \in \mathbb{R}^n : \left\|x\...
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3answers
103 views

Prove or disprove $f(x) = e^x$ is uniformly continuous on $(0, 1)$ [closed]

Prove or disprove $f(x) = e^x$ is uniformly continuous on $(0, 1)$. I know $f$ is uniformly continuous but how do I show this?
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2answers
40 views

$f: \mathbb{R}^{n+1}-\{0\}\rightarrow S^{n}$ via $x \mapsto\frac{x}{\|x\|}$ is continuous

I'm having trouble understanding why a map $f: \mathbb{R}^{n+1}-\{0\}\rightarrow S^{n}$ (unit $n$-sphere, $n\ge 1$) via $x \mapsto\frac{x}{||x\|}$ is continuous. Since Unit n-Sphere under Euclidean ...
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0answers
26 views

$f$ is uniformly continuous only if $g$ is constant

Let $g:\mathbb R\to\mathbb R$ be continuous and define $f:\mathbb R^2\to\mathbb R$ by $f(x_1,x_2)=g(x_1x_2)$. Show that $f$ is uniformly continuous only if $g$ is a constant function. I'm not sure ...
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Baby Rudin Exercise 4.13 Alternate Proof Verification

I would like to know if my proof of ex 4.13 is correct. Thanks! Exercise 4.13 in Rudin asks: Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function ...
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1answer
46 views

Is the function $f:\,x\mapsto\sin\left(2\pi x\lfloor x\rfloor\right)$ uniformly continuous? [closed]

The function $f,\,f(x)=\sin\left(2\pi x\lfloor x\rfloor\right)$ is uniformly continuous on the set of real numbers. Where $\lfloor x\rfloor$ is $\operatorname{floor}(x)$: $\operatorname{floor}(x)$ ...
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1answer
53 views

Trouble understanding uniform continuity

I recognize intuitively what it means for a function to be continuous (i.e. no jumps or breaks in the function), but the concept of being uniformly continuous seems to be over my head. I'm looking at ...
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1answer
161 views

Is $f(x)=\sum_{n=1}^\infty\frac{nx^2}{n^3+x^3}$ uniformly continuous on $[0,\infty)$?

Last week I had an assignment to show $f(x)=\sum\limits_{n=1}^\infty\frac{nx^2}{n^3+x^3}$ for $x\ge0$ does not converge uniformly, but I misread the question as "show $f(x)$ is not uniformly ...
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0answers
65 views

Equivalent definition of continuity, unif.continuity, equicontinuity, etc

I am trying to summarize some definitions regarding the different types of continuity I know, in my own words, and I would like to know if you think they are correct (that is, if they are equivalent ...
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2answers
116 views

Examples of (not) uniformly continuous, non-differentiable, non-periodic functions

Let $I\subseteq\mathbb{R}$ and $f:I\to\mathbb{R}.$ $(0)$ If $f$ is discontinuous on $I$, then it is not uniformly continuous. $(1)$ Suppose $I$ is open and bounded. If $f$ is unbounded on $I$,...
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3answers
147 views

Continuous function satisfying $f(x+1) = f(x)$

Let $f : \mathbb{R} \to \mathbb{R} $ be a continuous function and $f(x+1) = f(x)$ for all $x \in \mathbb{R}$. Then $f$ is bounded above, but not bounded below. $f$ is bounded above and ...
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1answer
52 views

Reverse Holder continuity

Consider a function $f(x)$ with a point-wise Holder exponent $\beta \leq 1$. Definition of point-wise Holder exponent: $$ \beta_x: = \sup \left\lbrace \beta: \limsup_{h \rightarrow 0^+} \left|\frac{...
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1answer
40 views

Lipschitz continuous and matrix

Suppose $f$ is Lipshitz continuous, that is $$\|f(t,u)-f(t,v)\|\le L^2\|u-v\|,$$ where $L^2$ is a Lipschitz constant. Why can we write this in the form: $$f^Tf\le L^2 x^Tx?$$
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1answer
17 views

A function taking values in a countable product of metric spaces is uniformly continuous iff its coordinate functions are.

Let $\lbrace (M_n,d_n) \rbrace _{n \in I}$ be a countable family of metric spaces and define a metric in the product $\prod _{n \in I}M_n$ as follows: $$d(\vec{x},\vec{y})=\sum _{n \in I} \min \lbrace ...
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0answers
25 views

How to prove a continuous function is uniformly continuous on a compact set using BW theorem?

Question. Let S be a compact set in $\mathbb R^n$ and $f:S\rightarrow \mathbb R^m$ be a continuous function. Prove that $f$ is uniformly continuous on S. I want to prove it using Bolzano-Weierstrass ...
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3answers
54 views

If $f(x) \to 0$ as $\|x\| \to \infty$ then $f$ is uniformly continuous

let $f: \mathbb R^n \to \mathbb R$ be continuous such that $f(x) \to 0$ as $\|x\| \to \infty$. how do I prove that $f$ is uniformly continuous on $\mathbb R^n$? ok I saw a question here where $n = 1$ ...
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0answers
17 views

$C_0$ function in one variable, compact domain in the other and joint continuity implies uniform continuity

Suppose we have two sets $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a continuous function $f:X\times Y\to \mathbb{R}$ such that, for each fixed $x\in X$, the function $f_x=f(x,\cdot)$ is ...
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1answer
92 views

prove that $|f(x)|\le k|x|$ for some constant $k$ [duplicate]

If $f:\mathbb R \to \mathbb R$ is uniformly continuous and $f(0)=0$ then prove that $|f(x)|\le k|x|$ for some constant $k$
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40 views

Uniform Continuity

This question has three parts. a) Difference between continuity and uniform continuity b) Geometrical meaning of uniform continuity c) Correct the example Definition of Continuity of a function ...
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2answers
107 views

If $g$ is uniformly continuous and $f(x)$ is close to $g(x)$ (for large enough $x$), is then $f$ uniformly continuous?

Suppose $f$ is continuous on $(0,\infty)$ and $f$ is simmilar $g$ for all $x>M$ ($M>0$). (i.e. for any $\epsilon >0$, there is $M>0$ such that if $x>M$, then $|f-g|< \epsilon$) is ...
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0answers
21 views

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\{ (0,0),(...
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14 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
46 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
33 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
50 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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1answer
79 views

Prob. 4 (a), Sec. 20 in Munkres' Topology, 2nd ed: Are these functions continuous in the product, uniform, and box topologies?

Here is Prob. 20 (a) in the book Topology by James R. Munkres, 2nd edition. Consider the product, uniform, and box topologies on $\mathbb{R}^\omega$. In which of these topologies are the ...
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2answers
28 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 $d(...
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0answers
48 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
48 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 $|\sin(t)| \leq 1$ forall $...
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1answer
22 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 f(x)-g(x)\...
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0answers
21 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
50 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
45 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
43 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 $\lim_{h\to0^{...
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1answer
45 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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43 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 $|x-...
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1answer
24 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 $\frac{\epsilon}...
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2answers
51 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$$$f_n(0)=0-0^n=0$$ $x\to \infty$ $$f_n(...
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2answers
102 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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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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34 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
45 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 ...
0
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1answer
51 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 $$\int_{\mathbb{R}}f(x)dx=M<\...
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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 $...
2
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0answers
45 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 ...