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".
352
questions with no upvoted or accepted answers
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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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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 ...
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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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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
$$
d_Y(f(x),f(y)) \le ...
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Baby Rudin Exercise 4.13 Alternate Proof Verification
I would like to know if my proof of ex 4.13 is correct. Thanks!
Exercise 4.13 in Rudin asks: Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function ...
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Groups where every continuous function is uniformly continuous
Let $G$ be a topological group, and $(X, d)$ a metric space. The function $f : G \to X$ is left uniformly continuous if for all $\varepsilon > 0$ there exists an identity neighbourhood $U$ such ...
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Conditional Expectation($E[f(X)\mid X \in S]$) is uniformly continous in its argument($S$)
I basically want to prove that under some conditions $E[f(X)\mid X \in S]$ is uniformly continuous in $S$
To be more specific, suppose $f:K \subset \mathbb{R}^n \rightarrow \mathbb{R}$ is a bounded ...
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Uniform continuity of $\frac{1}{2} \sin{ \frac{x}{2}}- \frac{1}{x}$
Let $f(x)=\frac{1}{2} \sin{ \frac{x}{2}}- \frac{1}{x}$ if $0<x \leqslant \pi$ and $f(x)=0$ if $x=0$.
I took the sequences $$x_n=\frac{1}{n}$$ $$y_n=\frac{1}{n^2}$$
We have that $x_n-y_n \...
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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 ...
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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 ...
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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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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)...
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1
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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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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
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0
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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||...
user999236
3
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1
answer
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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\...
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3
votes
0
answers
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(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 ...
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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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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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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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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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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 ...
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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 ...
user330531
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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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0
answers
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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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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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answers
28
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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}...
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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 ...
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3
answers
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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 \...
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0
answers
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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$ ...
user999236
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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)|\} |...
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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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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_{...
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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 ...
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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 ...
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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{...
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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 ...
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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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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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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 ...
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$\|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 ...
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0
answers
55
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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 = \...
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votes
1
answer
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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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0
answers
51
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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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votes
0
answers
34
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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
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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
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votes
0
answers
53
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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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40
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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 ...
user502136
2
votes
0
answers
50
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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 ) ...
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0
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$X$ 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 ; ...
user371663