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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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Real Analysis / Uniform Continuity [on hold]

Must a bounded continuous function on R be uniformly continuous? Justify your answer.
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1answer
46 views

$f(x)=x\cdot \ln(x)$ uniformly continuous in $(0,3]$? [on hold]

I have to decide if the function $f(x)=x\cdot \ln(x)$ in the interval $(0,3]$ is uniformly continuous but I don't know how to start. In general I have problems with this kind of proof. Please can ...
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1answer
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A uniformly continuous function with the supremum metric…

Question: Suppose that $C([0, 1])$ is the metric space of all continuous real-valued functions on $[0, 1]$, with the metric $d(f, g) := \sup_{x \in [0, 1]}|f(x) - g(x)|$. Let $f \in C([0, 1])$ such ...
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2answers
42 views

Let $f:K \rightarrow N$ be a continuous function from a compact $K$. Show that $f$ is uniformly continuous

I'm having trouble finishing this. One approach that I made is this: Let $\epsilon > 0$. Then, since $f$ is continuous, for every $x \in K$ exists $\delta_x > 0$ such that $d(x, x')<\delta_x ...
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47 views

If $f$ be a uniformly continuous function on $(a,b)$ then $f$ is bounded there.

We define a continuous extension of $f(x)$ to the set $[a,b]$, by $g(x)=f(x), x\in (a,b)$ and $g(a)= \lim_{x \to a^{+}}f(x)$ and $g(b)=\lim_{x \to b^{-}}f(x)$. $g(x)$ being continuous on a compact set ...
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1answer
60 views

How to compute the $n^{th}$ partial sum of a series?

Compute explicitly $S_n(x)$, the $n^{th}$ partial sum of the series $$\sum_{k=1}^∞ \frac{x\left[-1+4k(k+1)x^2\right]}{(1+4k^2x^2)(1+4(k+1)^2x^2)}$$ then compute the sum $S(x)$ of the infnite series, ...
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1answer
66 views

Prove that the function $f(x)=\frac{\sin(x^3)}{x}$ is uniformly continuous.

Continuous function in the interval $(0,\infty)$ $f(x)=\frac{\sin(x^3)}{x}$. To prove that the function is uniformly continuous. The function is clearly continuous. Now $|f(x)-f(y)|=|\frac{\sin(x^3)}{...
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0answers
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Integral uniform continuous? [duplicate]

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $\int_{-\infty}^{\infty} |f(t)|dt< \infty$, then $F(x) = \int_{-\infty}^{x} f(t)dt$ is uniformly continuous? My attempt: Since I ...
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1answer
39 views

Taylor's Theorem Remainder with Unbounded Derivative

According to the Wikipedia entry and a few I've seen online, the remainder form with a $(n+1) \text{th}$ derivative can be used as long as $f: \mathbb R \to \mathbb R$, is $n+1$ times differentiable ...
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1answer
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$f:[-1,1]\to\Bbb{R}$ be continuous function, and let $g(x)=\int_{0}^{1}f(xy)dy\forall x\in[-1,1]$

The whole question is Suppose $f:[-1,1]\to\Bbb{R}$ be continuous function and define $g(x)=\int_{0}^{1}f(xy)dy\ \forall x\in[-1,1]$. Prove that $g$ is continuous on $[-1,1]$. I have tried it in the ...
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1answer
26 views

Uniform continuity: $\delta < \epsilon$

I am reading a proof provided in Rudin's Principles of Mathematical Analysis regarding the Reimann-Stieltjes integral. In order to prove the statement (non relevant for the question itself) he ...
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1answer
59 views

$f_n(x)=n(f(x+\frac{1}{n})-f(x))$ converges uniformly to derivative

Let $f$ be a continuous function with continuous derivative on $(a,b)\subseteq\mathbb{R}$. Define $f_n(x)=n(f(x+\frac{1}{n})-f(x))$. Prove that $f_n$ converges uniformly to $f'$ on any ...
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2answers
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How do I have an intuitive understanding of modulus of continuity

I briefly chanced upon something called the modulus of continuity while starting on an introductory analysis course on limits and continuity: $\text{let } f:I \to \mathbb{R}$. Then for all $x,y \in I$...
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Confusion in the proof of Rouché's theorem (joint continuity part)

Let, $f,g$ be two holomorphic functions in a region $\Omega$. $C$ be a circle in $\Omega$ containing interior such that $|f(z)|>|g(z)|\ \forall z\in C$. g vanishes nowhere on $C$. Then $f$ and $...
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0answers
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To show continuity of a inverse distribution function

Let work with the space where distributions and its inverses exist. For any distribution $F$ its inverse for any $p\in(0,1)$ is $F^{-1}(p)=\inf \{t\in \mathbb{R} : F(t) \geq p \}.$ Continuity ...
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1answer
39 views

Uniform continuity of $x \sin{\frac{1}{x}}$ [duplicate]

$f(x) = x \sin{\frac{1}{x}}$ for $x > 0$, $0$ otherwise. Prove $f$ is uniformly continuous. I know similar questions have appeard asking for uniform continuity on $(0, 1)$, but here is uniform ...
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1answer
32 views

Limit of a function f(x) to infinity is 0 given f(x+n) to infinity is 0 for all rationals n in the interval (0,1)

I am given the following: $$ Let f:\mathbb{R} \to \mathbb{R} \text{ be uniformly continuous on } \mathbb{R}. \text{ Assume that for every rational } x \in (0,1), \lim_{n\to \infty}{f(n+x)} = 0. \text{...
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1answer
46 views

Multiple integrals and uniform continuity

For a function $$f (x,y)$$ continuous on $[a,b]\times J$, where $J$ is an open interval: $$g(y) = \int_a^b\ f(x,y)\ \mathrm dx$$ is also continuous. My question is: if $J$ is a closed then $f$ ...
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2answers
36 views

A question about compact support

Support is defined as the closure of $\{x:f(x)\neq 0\}$. Now consider $f(x)=\frac{1}{x}\chi_{(0,1)}$. So the support is $[0,1]$ which is compact. However, any continuous function with compact support ...
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1answer
53 views

Proof that a piecewise function is uniformly continuous

Let the function $f : \Bbb{R} \to \Bbb{R}$ be denoted by: $$f(x) = \begin{cases} \dfrac13 \sin(3x) + x^4 \sin\dfrac{1}{x^3}, \quad & x \ne 0 \\ 0, & x = 0 \end{cases}$$ Prove that $f$ is ...
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1answer
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$p(x)$ be a polynomial and $f(x)= \frac{1}{p(x)}$. Show that $f(x)$ is uniformly continuous.

Let $p(x)$ be a non-constant polynomial with real coefficients such that $p(x) \neq 0$, $\forall x \in \mathbb{R}$. Define $f(x)= \frac{1}{p(x)}$ for all real $x$. Prove that For each $\epsilon >0$...
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2answers
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Can I say $1/x^2$ is not uniformly continuous on $[0, \infty)$ because it is not defined when x = $0$?

This is in my homework to show if $1/x^2$ is uniformly continuous on $[0, \infty)$. I'm thinking since the definition of uniform continuity says for all $ x, y \in $ the domain, $|f(x) - f(y)| < \...
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1answer
73 views

Why coarse maps have to be proper?

A map $f: X \to Y$ between metric spaces is said to be coarse, if the following two conditions hold: $f$ is bornologous, i.e. $$\forall_{R>0} \; \exists_{S>0} \; d(x,y) < R \Rightarrow d(f(...
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1answer
62 views

Is $ f(x) = \log(1 + x^2)$ uniformly continuous on $\mathbb{R}$?

Is $f(x) = \log (1 +x^2) $ uniformly continious on $\mathbb{R}$? Yes/No. My attempt : yes Let $\Delta y:=x-y$, so that $x=y+\Delta y$. Then $$ \frac{1+x^2}{1+y^2}=\frac{1+y^2+2y\Delta y+(\Delta y)^...
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0answers
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Uniform Continuity on Compact Set with twist

I have been asked to prove the following: Have $g:[a,b]\rightarrow \mathbb{R}$ be continuous on $K = [a,b] \setminus \cup^\infty_{n=1} (\alpha_n,\beta_n)$. Then, for any $\epsilon > 0$, there is a ...
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1answer
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Let $f:(0,\infty)\to(0,\infty)$ be uniformly continuous function, is the following statement true?

Let $f:(0,\infty)\to(0,\infty)$ be uniformly continuous function. Does it imply $$\lim_{x\to\infty} {f(x+{1\over x})\over f(x)}=1\;?$$ By uniform continuity , for any $\epsilon>0$, $\exists\delta&...
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2answers
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How to prove, that $f\left(\bigcap\limits_{n=1}^\infty K_n\right)=\bigcap\limits_{n=1}^\infty f(K_n)$? [closed]

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function and $\{K_n\}_{n=1}^\infty$ be a decreasing sequence of compact subsets of $\mathbb{R}$. Show that $f\left(\bigcap\limits_{n=1}^\infty K_n\right)...
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1answer
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Is a piecewise linear function with uniformly bounded derivative uniformly continuous?

Let $(a_k)_{k \geq 0} \subset \Bbb R$ be any sequence. Let $f : \Bbb R \rightarrow \Bbb R$ be the continuous function defined by $$f(x) = \begin{cases} a_0 \text{ if } x \leq 0 \\ a_k + (a_{k+1}-a_k)...
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2answers
44 views

Real analysis; uniformly continuous function.

If $E$ is a non-compact and bounded set in $\mathbb{R}$, then there exists a continuous function on $E$ which is not uniformly continuous. In the proof, the book says that if $E$ is a non-compact and ...
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Continuity of a function on a metric space and its consequences

Let $f : X → Y$ be a given function, and suppose that $f^{-1}(C)$ is an open subset of $X$ whenever C is an open subset of $Y$ . (a) Prove that $f$ is continuous on $X$. (b) Prove that $f^{-1}(B)$ ...
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1answer
28 views

Proving a sequence is Cauchy from uniform continuity

Given: $f: X \longrightarrow Y$ is uniformly continuous on $X$, $(x_n)_n \in X $ is a Cauchy sequence. Question: What can you say about the sequence ${f(x_n)}$ ? My attempt: Since $f$ is uniformly ...
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2answers
412 views

Give an example of a function that is bounded and continuous on the interval [0, 1) but not uniformly continuous on this interval.

My thoughts was to take $f(x) =\cos(\frac 1x) $ for all $ x \in [0,1)$ as I know this function is continous from $[0,1)$ and is definitely not uniformly continuous as it oscilates non-uniformly. My ...
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1answer
42 views

Proving that a function is uniformly continuous having the limit [duplicate]

Let $$f:[0,\infty)\rightarrow \mathbb R$$ be a continuous function. Suppose that $$\lim_{x\rightarrow\infty}f(x) = L$$ Prove that f is uniformly continuous on $[0,\infty)$.
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Taylor series error?

I was playing around with Taylor series and a shifted Gamma Function and found something that doesn't work, and I'm not sure what I'm doing wrong. Is it some uniform continuity problem? Let: $$\Pi(n)...
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1answer
30 views

Continuity and Uniform Continuity of inverse functions

Let $f : X → Y$ be a given function, and suppose that $f^{-1}(C)$ is an open subset of $X$ whenever C is an open subset of $Y$ . (a) Prove that f is continuous on $X$. (b) Prove that $f^{-1}(B)$ is ...
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1answer
129 views

Every continuous function is uniformly continuous $\Rightarrow$ Lebesgue number exists for every open cover

How to prove the following proposition? Let $(X, d)$ be a metric space, if for any metric space $(Y,\tilde{d})$, any continuous map $f:(X,d)\rightarrow (Y,\tilde{d})$ is uniformly continuous then $(...
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1answer
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uniformly continuous imply bounded

Proposition: Let $(X,d)$ be compact metric space, and $Y$ be Borel subset of $X$. Suppose $A$ is homeomorophic to $Y$. Then, uniformly continuous function $f:A \to \mathbb{R}$ is bounded function. I ...
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0answers
27 views

Uniform continuity of a function on an interval

Let $h : (0, +\infty) \to \mathbb{R}$ be the function defined by $$h(x) = \cos(2x) + x\sin(1/x).$$ Prove that $h$ is uniformly continuous on $(0, +\infty)$. My attempt: $h$ is uniformly continuous ...
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1answer
25 views

Uniform continuity of addition of functions [closed]

If $f$ and $g$ are uniformly continuous functions on a set $E$, prove that the function $f + g$ is uniformly continuous on $E$.
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1answer
35 views

Continuity of a function and its consequences

Let f : X → Y be a given function, and suppose that $f^{−1}(C)$ is an open subset of X whenever C is an open subset of Y . Prove that f is continuous on X.
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3answers
38 views

Homeomorphic spaces are uniformly isomorphic

A continuous function $f$ is a homeomorphism if it is bijective, and open. A uniformly continuous function $f$ is a uniform isomorphism if it is bijective and $f^{-1}$ is uniformly continuous. Is ...
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1answer
30 views

Uniform Isomorphism

I have read somewhere that a function $f:(X,\mathcal{D}(X))\longrightarrow (Y,\mathcal{D}(Y))$ is a uniform isomorphism provided that $f$ and $f^{-1}$ are uniform continuous functions and $f$ is ...
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0answers
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Constant of continuity as a monotonically increasing function of the radius of the neighbourhood

While reading the fairly famous paper by Brezzi et al from 1980 (link), I got slightly puzzled by one of the conditions, in particular in Theorem 1 (Implicit Function Theorem basically). They consider ...
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1answer
45 views

Uniform inequality for a continuous function

Let $f(x,y)\in \mathcal{C}([a,b]\times[c,d])$ such that $$\exists \xi\in (a,b) : f(\xi,y)\neq 0, \forall y\in [c,d].$$ By the continuity of $f$, we have $$|f(\xi,\cdot)|\geq \min\limits_{[c,d]} |f(\...
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1answer
54 views

Continuity on infinity

We have $f: [a.b] \rightarrow \mathbb{R}$ continuous, and $$c_n = \sup\{c \in [a,b] : |f(x) - f(c_{n-1})| < \epsilon \text{ for any } x \in [c_{n-1},c]\}$$ with $c_1 = a$. In a previous exercise ...
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1answer
33 views

Rephrasing the “if” part of the statement of the Principle of Uniform Boundedness

I am trying to better and more rigorously understand the Principle of Uniform Boundedness (PUB). Recall that the statement of PUB reads: Let $X,Y$ be Banach Spaces and suppose that $(T_n)_{n\in\...
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1answer
87 views

Whether the product of uniformly continuous functions is uniformly continuous [closed]

I know it isn't and I have to give a counter-example. Function $f_1(x)=f_2(x)=x$ this is a uniformly continuous function the product of these functions $f_1(x)\cdot f_2(x)=x\cdot x=x^2$ this isn't an ...
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3answers
87 views

Log (uniform) continuous functions

I am interested in functions $f: \mathbb R_+\to \mathbb R_+$ such that $\log \circ f \circ \exp$ is uniformly continuous. In other words \begin{align} \forall_{c}\, \exists_{c'}, \forall_{ x,x'\in[...
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2answers
22 views

Showing |f(x)-f(y)| < $(\frac1\epsilon)^2*|x-y|$ for x,y on [$\epsilon^3/4,1$]

f(x) in this case equals $x^\frac13$ So far I've tried setting $|x-y| < \delta$, with $\delta$ = 2$\epsilon^3$, therefore making $(1/\epsilon^2)|x-y| < 2\epsilon$, but this doesn't show that ...
0
votes
1answer
28 views

Contraction and $\max$ function

$f: \Bbb R \mapsto \Bbb R$ $g: \Bbb R \mapsto \Bbb R$ $h: \Bbb R \mapsto \Bbb R$ $h:=\max\{f(x), g(x)\}$ Is $h$ a contraction on $ \Bbb R$ if $f$ and $g$ are both so? First attempts of ...