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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39 views

Show that if a function $f$ Lipschitz continuous on $X$, $f$ has to be uniformly continuous on $X$.

Show that if a function $f$ is Lipschitz continuous on $X$, $f$ has to be uniformly continuous on $X$. My attempt: (1) The definition of Lipschitz continuity for $f$ on $X$ is: $\exists L \in \...
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16 views

uniform continuity on disjoint intervals

Consider the following statements (a) If $f$ is uniformly continuous on disjoint closed intervals $I1,I2,......,In$, then $f$ is uniformly continuous on $\cup_{j=1}^n Ij$ (b) If $f$ is uniformly ...
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38 views

limit of $f:(0,\infty) \rightarrow \mathbb{R}$ as $x\to \infty$

Let $f:(0,\infty) \rightarrow \mathbb{R}$ be uniformly continous .Then $(1)\lim_{x\rightarrow 0+}f(x)$ and $\lim_{x\rightarrow \infty}f(x)$ exist $(2)\lim_{x\rightarrow 0+}f(x)$ exist but $\lim_{x\...
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18 views

uniformly continous function - upper limit by partition

The function $v:\Omega \subset \mathbb{R}^n_{x}\times \mathbb{R}_{t} \rightarrow \mathbb{R}^n$ is continous and has compact support. So it is even uniformly continous. From this property they follow ...
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1answer
36 views

Show that the direct sum $f\oplus g:X\to\textbf{R}^{2}$ defined by $f\oplus g(x) = (f(x),g(x))$ is uniformly continuous.

Let $(X,d_{X})$ be a metric space, and let $f:X\to\textbf{R}$ and $g:X\to\textbf{R}$ be uniformly continuous functions. Show that the direct sum $f\oplus g:X\to\textbf{R}^{2}$ defined by $f\oplus g(x) ...
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35 views

Let $f:X\to Y$ and $g:Y\to Z$ be two uniformly continuous functions. Show that $g\circ f:X\to Z$ is also uniformly continuous.

Let $(X,d_{X})$, $(Y,d_{Y})$ and $(Z,d_{Z})$ be metric spaces, and let $f:X\to Y$ and $g:Y\to Z$ be two uniformly continuous functions. Show that $g\circ f:X\to Z$ is also uniformly continuity. MY ...
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2answers
48 views

Value of a function that is uniformly convergent.

"Let $f$ be a function of $\mathbb{R}$ into $\mathbb{R}$ such that $\vert f(x)-f(y) \vert\leq\frac{\pi}{2}\vert x-y\vert^2$ for all $x,y\in\mathbb{R}$, and such that $f(0)=0$. What is $f(\pi)$?". ...
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1answer
40 views

Does this phenomenon regarding volumes of images of small balls hold uniformly?

Let $\Omega \subseteq \mathbb{R}^n$ be a nice domain with smooth boundary (say a ball), and let $f:\Omega \to \mathbb{R}^n$ be smooth. Set $\Omega_0=\{ x \in \Omega \, | \, \det df_x =0 \} $ In this ...
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1answer
27 views

Uniform Continuity of Characteristic Function

I am trying to understand the concept of uniform continuity as it pertains to characteristic functions. First my understanding of uniform continuity: Def: $$\forall x_0, \forall \epsilon>0, ...
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30 views

Prove that a function is uniformly continuous in $[a,\infty)$

Let $f:[a,\infty)\to\mathbb{R}$ be a continuous function. For every $\varepsilon>0$ there exist $0<\delta_{\varepsilon}$ and $a<c_{\varepsilon}\in\mathbb{R}$ so that for every $x_{1},x_{2}&...
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1answer
46 views

Proving uniformly continuity of 2 functions

Domain: $(0, \infty)$ I have $2$ functions: $$ f(x) = \sqrt{x}, \quad g(x) = x \cdot \sin(1/x) $$ The answers say that $f(x)$ is uniformly continuous because at $0$ it has a finite limit and in ...
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1answer
41 views

Lipschitz constant of continuous and piecewise linear functions

I want to calculate the Lipschitz constant of a continuous and piecewise linear function $f:[0,1]^2\rightarrow R$, like this \begin{equation*} f(x_1,x_2)=\left\{ \begin{aligned} 2x_1+x_2, &\quad\...
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1answer
54 views

$f(x,y)=\arcsin \frac{x}{y}$ is continuous but not uniformly continuous in its domain

I need to prove that $f(x,y)=\arcsin \frac{x}{y}$ is continuous, but not uniformly continuous on its domain. I noticed that the domain of the function is $D_f=\{(x,y)|-y\leq x \leq y$ if $y>0$, ...
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42 views

Let $f_n(x) = {nx\over 1+nx^2}$ on the domain $[-1,1]$

(a) Find the pointwise limit function $f$ on $[−1, 1]$. (b) Show that $\lim\limits_{n \to \infty} \int_{-1}^1 f_n(x)dx$ exists. Is it equal to $\int_{-1}^1 f(x)dx$? This is my solution: For part a, ...
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32 views

Prove the finite sum of bounded functions is bounded and similar result holds for continuous and uniformly continuous functions

Let $f_{1},f_{2},\ldots,f_{n}$ be a finite sequence of bounded functions from a metric space $(X,d_{X})$ to $\textbf{R}$. Show that $\sum_{i=1}^{n}f_{i}$ is also bounded. Prove a similar claim when ''...
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1answer
43 views

How to prove this function is not uniformly continuous?

I need to determine whether this function $f(x)=\log(2+\cos(e^x))$ is uniformly continuous on $\mathbb{R}$. I know this function is not uniformly continuous already from the graph of it, but I have no ...
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82 views

If $f$ is uniformly continuous on two open sets with a non-empty intersection, then $f$ is uniformly continuous on their union

There is a problem when I am solving this question:- Suppose $a<b<c<d$. Prove that if $f$ is uniformly continuous on $(a,b)$ and on $(c,d)$ then $f$ is uniformly continuous on $(a,b)\cup(c,...
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23 views

Prove that there exists a subsequence $\{g_{n_k}(x)\}$ converging uniformly to a continuous function $g(x)$ on $[0,1]$.

Let $f\in L^1[0,1]$, $E_n\subset[0,1]$ be measurable subsets and $$g_n(x)=\int_0^x\chi_{E_n}(t)f(t)\mathrm{d}t$$ where $\chi_{E_n}$ is the characteristic function of the set $E_n$. Prove that there ...
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3answers
42 views

(Uniform) continuity in $\mathbb{R^n}$

I have the following function $$ \mathbb{R^n} \setminus \{0\} \to \mathbb{R^n} : f(x) = \frac{x}{|x|^2} $$ equipped with the euclidean norm (p-norm with $p = 2$) I know I can analyze the continuity ...
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47 views

If $U_n\to U$ in probability, then for all countinuous function $L:\mathbb R\to \mathbb R$, $L(U_n)\to L(U)$ in probability. Proof unclear.

Let $(U_n)$ a sequence s.t. $U_n\to U$ in probability. Let $L:\mathbb R\to \mathbb R$ be a continuous function. Prove that $L(U_n)\to L(U)$. The proof goes as follow : Step 1 : Suppose $L$ is ...
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37 views

There exist a point $ c \in [a,b]$ such that $f(c)= \frac{f(x_{1})+f(x_{2})+…+f(x_{n})}{n}$.

Let $ f: [a,b]\to \mathbb{R}$ be continuous on $[a,b]$ and $ x_{1},x_{2},...,x_{n} \in [a,b].$Then there is a point $ c \in [a,b]$ such that $f(c)= \frac{f(x_{1})+f(x_{2})+...+f(x_{n})}{n}$. Can ...
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1answer
42 views

Proof that a continuous function on a closed interval is uniformly continuous

When proving that a function which is continuous on a closed interval $[a,b]$ is uniformly continuous, every proof is somewhat involved. Why does the following argument not suffice (or does it)? Since ...
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3answers
29 views

Uniform Continuity of a Piecewise Function

Suppose you have the function $f : [0,1] \rightarrow \mathbb{R}$, with $f(x) = 0$ if $x \in [0,1)$ and $f(x)=1$ if $x=1$. Prove that it is uniformly continuous. I got this function as the pointwise ...
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25 views

Prove that $f:X\rightarrow\Bbb{R}$ is uniformly continuous iff it maps equivalent sequences onto equivalent sequences

Let $X$ be a subset of $\Bbb{R}$, and let $f:X\to\Bbb{R}$ be a function. Then the following two statements are logically equivalent: (a) $f$ is uniformly continuous on $X$. (b) Whenever $(x_{n})_{n=...
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30 views

Uniform convergence - Is f continuous and is f differentiable $f(x) = \sum_{n=1}^{\infty}\frac{1}{2n^2-\sin(nx)}$

Let f(x) = $\sum_{n=1}^{\infty}$$\frac{1}{2n^2-\sin(nx)}$ ($x\in\mathbb{R}$) (a) Decide whether f is continuous on R. (b) Is f differentiable? Don't even know where to begin with this question, I ...
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2answers
27 views

Proving non-uniform continuity of functions.

Say we wanted to show that $f(x)=\frac{1}{x}$ was not uniformly continuous on $(0,1)$, I will restate a proof I saw on another question, or rather a hint that I saw and my attempt to formulate a proof ...
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1answer
101 views

$g(x) = f(x)\sin\ (1/x)$ being uniformly continuous on $(0, 1]$

Let $f: (0, 1]\to \mathbb{R}$ be continuous on the domain. I want the condition of $f(x)$ where $g(x) = f(x)\sin(1/x)$ is uniformly continuous on $(0, 1]$. I expect that the answer would be $\lim_{x\...
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26 views

prove that function $f(x)=\frac{\sin(x^2)}{\sqrt{x}}$ is uniformly continuous by $\epsilon-\delta$

Instruction in book: "prove that for all $\epsilon > 0$ exists $M > 0$ s.t. $|f(x)| < \frac{\epsilon}{2}$ and on interval $[0,M]$ apply Cantor's theorem(if function is continuous on bounded ...
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1answer
14 views

Example of the two uniformly equivalent metrics, one is bounded while another is not.

Can anyone gives me an example such that two metric space (X,d1), (X,d2) are uniformly equivalent. And the metric space (X,d1) is bounded and metric space (X,d2) is not bounded?
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16 views

Some questions about the proof of the properties of the uniform equivalent.

I want to prove some properties of uniformly equivalent metrics. Suppose (X,d) and (X,p) are uniformly equivalent, then the identity map and its inverse are uniformly continuous. The former is ...
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24 views

How to prove the uniform equivalence is indeed an equivalence relation on the class of metrics on X

May I ask a homework question? I'm just wandering the equivalence relation is defined on two sets while the uniformly equivalent is defined on two metrics. How can they be equal? And how to prove that?...
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3answers
88 views

Continuity implies uniform continuity

What shown below is a reference from "Analysis on manifolds" by James R. Munkres First of all I desire discuss the compactness of $\Delta$: infact strangerly I proved the compactness of $\Delta$ in ...
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32 views

$f>0$ uniformly continuous and $\int_0^{\infty} f(x) \,dx \leq M$ imply $\lim_{x \to \infty} f(x)=0$?

I know that if $f$ uniformly continuous and $\int_0^{\infty} f(x) \,dx = c$, then $\lim_{x \to \infty} f(x)=0$ (link: $f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \...
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59 views

Is “bounded” and “Cauchy-continuous” function uniformly continuous?

Is a "bounded" and "Cauchy-continuous" function uniformly continuous? I have found lots of questions that ask whether "bounded" and "continuous" function is uniformly continuous. (I know the answer is ...
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1answer
23 views

Cauchy sequence of a uniformly continuous image

If $\{fx_n\}$ is a Cauchy sequence with $f$, a continuous self map on a complete metric space, we know that $\{x_n\}$ need not be Cauchy. Is it true for a uniformly continuous $f$? Here's my take: ...
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1answer
41 views

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
34 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
52 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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24 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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1answer
22 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
59 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
99 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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1answer
30 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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2answers
29 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
49 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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50 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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24 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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2answers
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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