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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9 votes
1 answer
173 views

What kind of "geometric" regularity $f'^2$ gives on $f$

When solving real-analysis' problems I like to represent the functions involved and think geometrically what is going on. Today I got the following exercise : Let $f \in \mathcal{C}^1(\mathbb{R},\...
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8 votes
0 answers
541 views

A sequence of functions that is uniformly continuous, pointwise equicontinuous, but not uniformly equicontinuous when their domain is noncompact

I'm trying to prove my sequence of functions $(f_n) = \frac{n}{n+1}\cos(x^2)$ on (0,1) is pointwise equicontinuous, uniformly continuous, but not uniformly equicontinuous. But, I'm having a lot of ...
5 votes
0 answers
94 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 ...
5 votes
0 answers
102 views

Functions $f: \mathbb{R}^n \to \mathbb{R}$ such that $|f(x) -f(y)| \le C \prod_{i=1}^n |x_i - y_i|^{\alpha_i}$

The standard definition of a Holder continuous function between metric spaces $X,Y$ is a function $f: X \to Y$ such that there exist $C>0$ and $0 < \alpha \le 1$ such that $$ d_Y(f(x),f(y)) \le ...
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5 votes
0 answers
383 views

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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4 votes
0 answers
80 views

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 ...
  • 2,855
4 votes
0 answers
83 views

Conditional Expectation($E[f(X)\mid X \in S]$) is uniformly continous in its argument($S$)

I basically want to prove that under some conditions $E[f(X)\mid X \in S]$ is uniformly continuous in $S$ To be more specific, suppose $f:K \subset \mathbb{R}^n \rightarrow \mathbb{R}$ is a bounded ...
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4 votes
0 answers
55 views

Uniform continuity of $\frac{1}{2} \sin{ \frac{x}{2}}- \frac{1}{x}$

Let $f(x)=\frac{1}{2} \sin{ \frac{x}{2}}- \frac{1}{x}$ if $0<x \leqslant \pi$ and $f(x)=0$ if $x=0$. I took the sequences $$x_n=\frac{1}{n}$$ $$y_n=\frac{1}{n^2}$$ We have that $x_n-y_n \...
4 votes
0 answers
463 views

In what sense is metric space completion universal?

The completion of a metric space is unique up to metric monomorphism (usually called isometry). It is also the "obvious" way to make all Cauchy sequences convergent. Structures which are unique up ...
4 votes
0 answers
555 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 in ...
4 votes
1 answer
136 views

Is this an alternate way to check if a function is uniformly continuous?

My question is whether the following statement is true, or if there exists one similar. For a differentiable real function $f:S\rightarrow \mathbb{R}$ where $S$ is an interval, and a fixed $c \in S$, ...
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4 votes
0 answers
98 views

Find a constant $C_p$ that satisfies $|f(x+p)-f(y+p)|\le C_p|f(x)-f(y)|$

Let $B^n$ be the unit open ball in $\mathbb{R}^n$, $p\in \mathbb{R}^n$ and $f\colon \mathbb{R}^n\to B^n$ defined as $f(x)=\frac{1}{1+|x|}x$. I believe there are constants $C_p>0$ such that $|f(x+p)...
3 votes
1 answer
1k views

problem about uniform continuity

Problem: "Show that if $f:\left[a,+\infty\right] \rightarrow \mathbb R$ is continuous and if $\lim_{x \rightarrow + \infty} f(x)$ exists, then $f$ is uniformly continuous". I have a question about ...
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3 votes
0 answers
88 views

Example of a compact operator that is not uniformly continuous. [Solved]

I want to find a Banach space $E$ and a compact operator $K:[0,1]\times E \rightarrow E$ (that is, $K$ maps every bounded sequence onto a sequence that converges up to a subsequence) satisfying the ...
3 votes
0 answers
63 views

Proving $f:[-1,1] \to \mathbb{R}$ with $f(x)=x^3$ is uniformly continuous

Can anyone verify the steps in my proof are correct? Scratch work: $|f(x)-f(y)| =|x^3-y^3|=|x-y||x^2+xy+y^2|$ (at this point note that we already know $|x-y|<\delta$ for some $\delta>0$) $$|x-y||...
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3 votes
1 answer
54 views

Is $T$ uniformly continous?

Let $(X,\lVert \cdot\lVert)$ be a normed space. Let $T$ be a function given by \begin{equation} \begin{array}{cccc} T:&X&\longrightarrow&X\\ &x&\longmapsto&T(x)=x\lVert x\...
3 votes
0 answers
91 views

(Uniform) convergence rate of fourier series under Holder-continuity

Let $f \in C^{\alpha}(\mathbb{T})$ be a holder-continuous function, i.e. $$ |f(x) - f(y)| \le C|x-y|^{\alpha} $$ for some $0 < \alpha < 1$. We use $\|f\|_{C_\alpha}$ to denote the smallest ...
3 votes
0 answers
65 views

Continuity of $f$ and monotone decreasing of $g(x) = \cos(f(x))$ implies uniform continuity of $f$

Let $f,g : \mathbb{R} \rightarrow \mathbb{R}$. I want to show uniformly continuity of $f$ under $f$ is continuous and $f(0) = \frac{\pi}{2}$, $g(x) = \cos(f(x))$ is monotonically decreasing. I know ...
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3 votes
0 answers
72 views

If $f(x) = x^2$ is uniformly continuous on open set A, then is $f$ bounded?

Let $A$ be a nonempty set of real numbers and let $f : A \rightarrow [0,\infty)$ be given by $f(x) = x ^2$. Prove or disprove that if $A$ is open and $f$ is uniformly continuous, then $A$ is bounded. ...
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3 votes
1 answer
476 views

Conditions for a function to be Lipschitz or just uniformly continuous

Question. Let $\Omega$ be an open subset of $\mathbb R^3$, and let $f: \Omega \to \mathbb R$ be continuous. Then $f$ is (a) Lipschitz, if it is $C^1$ and $\nabla f$ is bounded; (b) uniformly ...
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3 votes
0 answers
64 views

Showing uniform convergence of a sequence of functions which are inherited from a different function

Let $f: \Bbb R \times [0,1] \to \Bbb R$ be a continuous function and $(x_n)$ be a sequence in $\Bbb R$ that converges to $x$ in $\Bbb R$. Then show that the sequence of functions $g_n$ defined by,$$...
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3 votes
0 answers
239 views

Is a Function on Bounded Domain Taking Cauchy Sequences to Cauchy Sequences Uniformly Continuous?

I have a question similar to this one: Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous? So, from that discussion, we know that a function that takes Cauchy sequences ...
3 votes
0 answers
1k 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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3 votes
0 answers
107 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 $[a, b]$...
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3 votes
0 answers
50 views

Convergence in the product of spaces of iteratively composed functions.

My question is a bit odd, in fact conceptually it is not difficult, only that it operates on objects that are complex (to me). I would like to check two types of convergence in the product of the ...
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3 votes
0 answers
683 views

Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let $...
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2 votes
0 answers
28 views

STFT uniform continuity

I want to show that short-time Fourier transform of $f \in L^2$ w.r.t $g \in L^2$ \begin{align*} \mathcal{V}_{g} f (x, \omega) = \int\limits_{\mathbb{R}}^{} f(t) \overline{g(t-x)} e^{-2 \pi i t \omega}...
2 votes
0 answers
70 views

What is the sence of "$f(x,y)$ is continuous at $x_0$ uniformly in $y$"?

Consider a function $f(x,y): \mathbb{R}^2 \to \mathbb{R}$. I found the following phrase in a book: "$f(x,y)$ is continuous at $x_0$ uniformly in $y$". What does it mean? I think that it ...
2 votes
3 answers
100 views

Uniform continuity of the function $f(x)=x^a$

Prove that $f:[0, \infty ) \to [0, \infty)$, $f(x)=x^a$ with $a>0$ is uniformally continous if and only if $0<a\leq 1$. I guess that we could start with the scratch work like this: $\forall \...
2 votes
0 answers
40 views

Proving $1/x$ is uniformly continuous on $[1/2,1]$

We wish to prove that $f(x)=\frac{1}{x}$ defined on $[1/2,1]$ is uniformly continuous. Scratch work: Take $x,y \in [1/2,1]$, $|f(y)-f(x)|=|\frac{1}{y}-\frac{1}{x}|=|\frac{y-x}{xy}|$ At most $x=1,y=1$ ...
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2 votes
0 answers
48 views

Proof of uniform continuity of $f(x) = \inf\{\mid x - a \mid : a \in S\}, S \subset \mathbb{R}$

I was wondering if my proof of the uniform continuity of $f(x) = \inf\{| x - a | \colon a \in S\}, S \subset \mathbb{R}$ was correct. Attempt: $ | f(x) - f(y) | = | \inf\{|(x-a)|\} - \inf\{|(y-a)|\} |...
2 votes
0 answers
51 views

Prove sin(x) is uniformly continuous on R with Heine-Cantor Theorem

I've already proven uniform continuity of f(x)=sin(x) on R via epsilon-delta, but I wanted to try and prove it with the Heine-Cantor-Theorem since it seems more intuitive: Now obviously with the ...
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2 votes
0 answers
32 views

Is the follwoing function an example of an uniformly continuous function?

Is $id$ an uniformly continuous function? Let $(\mathbb{N},d)$ be a metric space such that $d(x,y) = |\frac{1}{x}-\frac{1}{y}|$. Let us consider the identity map $id:(\mathbb{N},d) \to (\mathbb{N},d_{...
  • 3,392
2 votes
0 answers
51 views

Alternative proof that a continuous function on metric spaces with compact domain is uniformly continuous

I just came up with the following proof. What I like about it is that when I came up with it I felt like I was just following my nose. It's somewhat different from the other ones I've seen, so I'd ...
  • 1,138
2 votes
0 answers
64 views

Non-trivial metric space $X$ with all of $C(X)$ uniformly continuous.

Is there a metric space $X$ such that any $f\in C(X)$ is uniformly continuous, but there is a metric space $Y$ and $g:X\to Y$ which is continuous but not uniformly continuous? If $X$ is compact or ...
  • 6,192
2 votes
0 answers
65 views

What properties does this function have?

Let $f\colon\mathbb{R} \rightarrow \mathbb{R}$ be an $\textbf{uniformly continuous}$ function. Then by definition $\forall\epsilon>0 \quad \exists\delta>0\quad$such that $\forall x,y\in\mathbb{...
  • 103
2 votes
0 answers
81 views

Uniformly continuous $f$ such that $\lim_{n \to +\infty}f(nx) = 0 \implies \lim_{x \to +\infty} f(x)= 0$

Let $f : \mathbb R \to \mathbb R$ a uniformly continuous function such that $\forall x > 0$ $$\lim_{n \to +\infty}f(nx) = 0$$Show that $$\lim_{x \to +\infty} f(x)= 0$$ I cannot use Baire Category ...
  • 514
2 votes
0 answers
46 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 ...
  • 1,087
2 votes
0 answers
65 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=...
  • 21.4k
2 votes
0 answers
51 views

Does the limit of this integrand (product of monotonic functions) exist, given the integral is bounded?

For $p$, $f$ sufficiently differentiable functions: \begin{equation} \int_{-\infty}^{\infty} f^2 p = 1 \qquad \text{(i.e. $f \in L^2_p$)} \end{equation} I want to conclude $\lim_{x\to\pm\infty} f^2 p ...
  • 391
2 votes
0 answers
28 views

$\|T_af-f\|_\infty\rightarrow 0,\ a\rightarrow 0\Rightarrow\ f$ has uniformly continuous representative

Let $f\in L^\infty(\mathbb{R}^d)$ (where the measure we're working with is the Lebesgue measure) and I have to prove the following equivalence : $f$ has a uniformly continuous representative if and ...
  • 996
2 votes
0 answers
55 views

Proving uniform continuity for $x/(1+x)$ on $[0,\infty)$

I want to show that $f(x) = x/(1+x)$ is uniform continuous on $[0, \infty)$. My proof: Let $x,y$ be real numbers in $[0, \infty)$ and let $\epsilon$ be a positive number. We choose $\delta = \...
  • 169
2 votes
1 answer
174 views

Definition of uniformity in different contexts

My question is rather a semantic one. I am wondering what uniformity in different contexts in general means. I know what the definition of uniform continuity is. Has uniformity in general something ...
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2 votes
0 answers
51 views

Is $f:\mathbb{R}^{2} \backslash E \times\{0\}\to\mathbb{R}$ uniform continious function?

Can you help me with this problem? Let $E\subseteq \mathbb{R}$, $U=\mathbb{R}^{2} \backslash E \times\{0\}$. Prove that the following statements are equivalent: 1)Any continious function $f:U\to\...
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2 votes
0 answers
34 views

Extraneous Condition in the Hypothesis?

If $\{f_n\}$ is a sequence of continuously differentiable functions on $[a,b]$, $f_n \to f$ uniformly on $[a,b[$, and there is a function $g : [a,b] \to \Bbb{R}$ such that $f'_n \to g$ uniformly on $[...
  • 7,370
2 votes
0 answers
148 views

Prove that, any continuous and periodic function on $\Bbb{R}$ is uniformly continuous on $\Bbb{R}$.

There is a hint in my book to solve the problem. It goes as follows- Suppose, $f:\Bbb{R}\to\Bbb{R}$ is continuous and has the period $p\in\Bbb{R}$. Hence, $f(x+np)=f(x)\forall x\in\Bbb{R}, \: \...
  • 2,673
2 votes
0 answers
53 views

Reference for theorem bounding distances of values of uniformly continuous functions?

I think I proved the following theorem, but the proof is pretty long. So I thought maybe there is a reference for it because the statement seems standard. Does anyone know such a reference? Thank you! ...
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2 votes
0 answers
40 views

Existence of smallest $\delta>0$ for a continuous function

There's something on my textbook that I don't understand: Let $f:I \to \mathbb{R}$ be continuous on the interval $I$.So $\forall x_0$ $\epsilon$ $I$ and $\forall \epsilon>0$ there $\exists ...
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2 votes
0 answers
50 views

Uniformly Convergent Series 1 Fourier Coefficients

Let $f$ be an absolutely integrable function on $(0,2 \pi)$. We have $F(x) := \int_0^x f(t) \, dt$, $$F(x)= \frac{\alpha_0}{2} x + \sum_{k=1}^\infty \frac{\alpha_k \sin k x + \beta_k ( 1 - \cos k x ) ...
  • 9,110
2 votes
0 answers
265 views

$X$ a connected space ; if all continuous functions from $X$ to $\Bbb R$ are uniformly continuous, then X is a compact

Let $X$ be a subset of a normed vector space. $X$ is also a connected space. Show that if all continuous functions from $X$ to $\Bbb R$ are uniformly continuous, then X is a compact. Need some help ; ...
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