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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On uniformly Lipschitz function.

Let $f:[t_{0}, T]\times X\to X$ be uniformly Lipschitz in $X$, what does uniformly Lipschitz function mean for this case? This question arises from the Theorem 1.7 which appears in the image. Thanks ...
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28 views

Sequential Criterion for the absence of Uniform Continuityy

I am reading Abbott's Understanding Analysis wherein the sequential criterion for the absence of uniform continuity is given as follows: A function $f : A \to\mathbb R$ fails to be uniformly ...
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continous functions on compact metric spaces are uniformly continuous [closed]

While reading about continuous functions on compact metric spaces , we study a theorem that a continuous function on a compact metric space is uniformly continuous. While going through the proof after ...
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Groups where every continuous function is uniformly continuous

Let $G$ be a topological group, and $(X, d)$ a metric space. The function $f : G \to X$ is left uniformly continuous if for all $\varepsilon > 0$ there exists an identity neighbourhood $U$ such ...
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23 views

Proof that a continuous function on a compact set is uniformly continuous, compact set not containing infimum

I'm reviewing the proof of the following statement (Baby Rudin 4.19): Let $f$ be a continuous mapping of a compact metric space $X$ into a metric space $Y$. Then $f$ is uniformly continuous on $X$. ...
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Proving that $f(x) = \frac{1}{x}$ is not uniformly continuous over $(0,1)$ - Approach to choosing the correct $x$ and $y$

I've looked at the various other posts on this question and I've also worked on uniform continuity in the past. But right now as I work through Spivak's Calculus, I'm seeing that I haven't actually ...
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Help me in understanding Heine-Borel theorem's proof

I am trying to understand the proof of Heine-Borel theorem given here. If I understand the meaning behind the phrases in box, I think it is enough for me to understand the proof. Please explain the ...
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Uniform continuity of a function: multidimensional distribution function

If a distribution function $F$ defined on $\mathbb{R},$ is continuous then it's uniformly continuous, this is easy to prove, since more generally if $f$ is continuous on $\mathbb{R}$ and has finite ...
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$f:\mathbb{R} \to \mathbb{R}$ is differentiable function such that $f'$ is bounded .Is $f+f'$ uniform continuous?? [closed]

$f:\mathbb{R} \to \mathbb{R}$ is differentiable function such that $f'$ is bounded, then we know $f$ is uniform continuous. But is $f+f'$ uniform continuous? I think no but I am not able to find a ...
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Check function uniform continuity [closed]

The task is to check function uniform continuity in terms of the following set $\{(x,y): x^2+y^2 \geq 2\}$: $$f(x,y)=(x^2+y^2)\cdot \sin\left(\frac{1}{x^2+y^2}\right)$$ Can you help me with this one? ...
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Show that the inequality $|f(x)-f(y)| \leq M||x-y||+\epsilon$.

Suppose $K \subset \mathbb{R}^n$ is a compact set and $f:K \rightarrow \mathbb{R}$ is continuous. Let $\epsilon >0$ be given. Prove that there exists a positive number $M$ such that for all $x$ and ...
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65 views

Proving that $\sin(x+y)$ is uniformly continuous

How may I prove that both are uniformly continuous?: $f(x,y)=\sin(x+y)$ for 1, I said, let $\epsilon > 0$ if $d((x,y),(u,v))<\lambda$: $|f(x,y)-f(u,v)|<|f(x,y)|+|f(u,v)|=|\sin(x+y)|+|\sin(u,...
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Bounded partial derivatives on convex set implies uniform continuity

Let $f(x,y):\mathbb{R^2} \rightarrow \mathbb{R}$, where $A \subset \mathbb{R^2}$ is convex, $f_x, f_y$ are bounded on $A$. How does one actually show that $f$ is uniformly continuos on $A$? I imagine ...
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Prove $f(x) = \sqrt{x}\ln(x)$ is uniformly continuous for $x = [1, \infty)$

Original question is to show this is true for all $x > 0$ with the hint to split cases on $x \in (0,1]$ and $x \in [1, \infty)$. I can show this is true for $x \in (0,1]$ by extending the interval ...
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Is $3x^2+7$ uniformly continuous on $[0,1]$?

Is $3x^2+7$ uniformly continuous on $[0,1]$? My argument: Yes, Since $3x^2+7$ is continuous on $[0,1]$ and $[0,1]$ is compact, any continuous function on compact set is uniformly continuous. Am I ...
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1answer
51 views

If $f(t)$ converges as $t \to \infty$ and $f'(t)$ is uniformly continuous then $f'(t) \to 0$ as $t \to \infty$ where $t$ is time.

I came across this result while I was reading mathematical epidemiology. What I interpreted from the statement of the result is that if a function is ultimately becoming constant then it's derivative ...
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uniformly continuous function is closed and bounded space

Important Note (before closing this as duplicate): I saw multiple solutions to this problem but all of them using balls which I don't want. I need to prove that: If $f : D \to \Bbb R$ is a ...
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1answer
21 views

Uniform continuity implies continuity in topological vector spaces

Let $E$ and $F$ be topological vector spaces and $A \subset E$. I want to prove that: if $f: A \longrightarrow F$ is a uniformly continuous function, then $f$ is continuous. I want that, by ...
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41 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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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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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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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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59 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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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
45 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
29 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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34 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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47 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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49 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
55 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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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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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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54 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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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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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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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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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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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
64 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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36 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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51 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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28 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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102 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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17 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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22 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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28 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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96 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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38 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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62 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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