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".
2,032
questions
0
votes
0
answers
17
views
Checking that topological group has uniform structure.
By definition of the right uniformity:
$V\in\Phi_R \Leftrightarrow \exists M\in\mathcal{N}_{\mathcal{T}_G}(0_G):\{(x,y)\in G\times G:x\cdot y^{-1}\in M\}\subseteq V$
During the proving that $\Phi_R$ ...
- 760
2
votes
1
answer
39
views
The map $f*g$ is uniformly continuous
Let $p, q \in [1, \infty]$ such that $\frac{1}{p}+\frac{1}{q} = 1$. We define the convolution operator
$$
* : L^{p} (\mathbb R^d) \times L^{q} (\mathbb R^d) \to L^\infty (\mathbb R^d)
$$
by
$$
(f*g) (...
- 13.8k
1
vote
0
answers
26
views
Uniform convergence, Pointwise Equicontinuity and Uniform Equicontinuity for functon.
I am trying to answer the following question:
Define $f_n(s)=s^n$ where $f_n:[0,1]\rightarrow\mathbb{R}$. Is the family of functions $\{f_n\}$:
Uniformly convergent
Pointwise Equicontinuous
...
- 91
0
votes
0
answers
50
views
$A\subseteq X$ is compact and $f:A\subseteq X\rightarrow Y$ is continuous. Show $f$ is uniformly continuous. [duplicate]
Theorem. Let $(X,d)$ and $(Y,\rho)$ be metric spaces, $A\subseteq X$ compact and $f:A\subseteq X\rightarrow Y$ continuous. Then $f$ is uniformly continuous.
Proof.
We have to prove that $f$ is ...
- 1
3
votes
2
answers
98
views
Asking for clarification of a Wikipedia article on uniform continuity
A portion of Wikipedia article on uniform continuity
Functions that have slopes that become unbounded on an infinite domain cannot be uniformly continuous. The exponential function $ x\mapsto e^{x}$ ...
- 11.7k
0
votes
1
answer
33
views
Counterexample to uniformly continuous extension theorem when A is not dense in X
We have that if $X$ is a metric space, $Y$ is a complete metric space, $A\subseteq X$ is dense in $X$ and $f:A \to Y$ is uniformly continuous then there exists a (unique) uniformly continuous ...
- 67
0
votes
2
answers
62
views
"Algorithmically", what does uniform continuity means?
I am a bit confused trying to understand what uniform continuity means, from the definition we have:
$$\forall \varepsilon > 0 \; \exists \delta > 0 \; \forall x \in X \; \forall y \in X : \, ...
- 23.2k
0
votes
4
answers
52
views
Justify the interchange of limits for $\displaystyle\lim_{n \to \infty} \lim_{m \to \infty} (n+1) \int_0^1 x^nP_m(x)dx$ where $P_m(x)$ is a polynomial
I am trying to prove the following problem:
Let $f(x)$ be a real and continuous function on $[0,1]$. Show that $$\displaystyle\lim_{n \to \infty} (n+1) \int_0^1 x^n f(x) dx = f(1).$$
I have shown ...
- 2,981
0
votes
2
answers
166
views
Suppose $f:\mathbb{R} \to \mathbb{R}$ is uniformly continuous. Show that $f(x+1)-f(x)$ is bounded
Suppose $f:\mathbb{R} \to \mathbb{R}$ is uniformly continuous. Show that $f(x+1)-f(x)$ is bounded.
I have considered the question, and my current approach is to show that there exists $\delta > 1$ ...
- 93
0
votes
0
answers
26
views
Uniform continuity of characteristic function of a tight family of measure
I am missing a step in the proof of Theorem 15.22 of Probabiliy Theory by A. Klenke (3rd version).
The theorem states that, given a tight family of probability measure on $\mathbb{R}$, the family of ...
- 193
0
votes
1
answer
39
views
If $f:[0, t] \to \mathbb R$ is continuous and has bounded variation, then its quadratic variation is $0$
Let $f:[0, t] \to \mathbb R$. For $n \in \mathbb N$, we define
$$
\begin{align}
V^{(1)}_{n} &:= \sum_{i=1}^{2^n} \left|f\left(\frac{i t}{2^n}\right)-f\left(\frac{(i-1) t}{2^n}\right)\right|, \\
V^{...
- 3,914
2
votes
1
answer
103
views
Prove function $f:\mathbb{R} \rightarrow \mathbb{R}$, $f(x)=e^{x^2}$ is not uniformly continuous.
I try this proof by contradiction that
Assuming $f:\mathbb{R} \rightarrow \mathbb{R}$, $f(x)=e^{x^2}$ is uniformly continuous.
By definition we have $\forall \varepsilon>0$ $\exists \delta>0$ $|...
- 21
1
vote
1
answer
87
views
if $f(x)x^2$ is uniformly continuous than limit of $f(x)$ is equal to $0$ [duplicate]
Let $f(x): \mathbb{R} \rightarrow \mathbb{R}$ be such that $g(x) =f(x)\times x^2$ is uniformly continuous. Prove that $\lim_{x\to\pm\infty}f(x)=0$.
Since $g$ is uniformly continuous than for every $\...
- 11
1
vote
2
answers
53
views
Integration over a Translated Set is Uniformly Continuous?
I'm looking for help with the following problem (working with Lebesgue measure here):
Let $ψ∈L^\infty(R)$ and let $B \subseteq \mathbb R $ be measurable such that $m(B) < \infty$. If $B+x =\{b+x|b∈...
- 11
1
vote
1
answer
40
views
Show uniform continuity of $x\longmapsto\sqrt[4]{\lvert x\rvert}$
I would like to show that $f\colon\mathbb{R}\to\mathbb{R}, x\longmapsto\lvert x\rvert^{1/4}$ is uniformly continuous.
My idea is the following. Let $\varepsilon>0$ and set $\delta=\varepsilon^4$. ...
- 153
2
votes
0
answers
32
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}...
- 893
0
votes
0
answers
20
views
proving that a function that's continous on $[a, +\infty$) with an horizontal or oblique asymptote (at +$\infty$) is uniformally continious [duplicate]
here's how I tried this (I guess it must be wrong),
let's firstly prove the part about the horizontal one,
if there exists an horizontal asymptote $\implies \lim_{x\to +\infty}f(x)=k\in \mathbb{R} $ ...
0
votes
0
answers
26
views
Let $f: [0,\infty) →\mathbb{R}$ uniformly continuous such that $\lim_{n \to +\infty} f(x+n)=0\; \forall x\geq 0$: show that $f(x) =0$ as $x\to\infty$ [duplicate]
To me it seems clear since its valid for any $x \geq 0$ so just take any $x$ greater than an given $n$ for which $|f(n)| < \varepsilon$ for some $n > N$. I can't see why uniform continuity is ...
2
votes
0
answers
71
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 ...
- 550
-3
votes
1
answer
53
views
Is the function $f(x) = \frac{1}{x}$ uniformly continuous on the interval $(0.2 ,1)$? [closed]
Is the function $f(x) = \frac{1}{x}$ uniformly continuous on the interval $(0.2, 1)$?
edit1:
I know from my textbook and other posts that the function $f(x) = \frac{1}{x}$ is not uniformly continuous ...
- 1
4
votes
1
answer
162
views
Proving $f(x) = x^2$ is not uniformly continuous on the real ine [duplicate]
I am aware that it is a duplicate of this post, but I haven't seen anyone presenting the proof which I used.
Could you kindly verify whether my proof is valid please?
The intuition is as follows:
If a ...
- 237
-1
votes
1
answer
66
views
An exercise I made for myself on Uniform Continuity and couldn't solve
I'm trying to learn Uniform Continuity in Real Analysis, so I made myself some exercises to solve and improve on $\epsilon-\delta$ proofs, but I couldn't solve one of the exercises:
"Prove that ...
- 195
2
votes
1
answer
74
views
Let $(X_n)$ be a sequence of $\text{Unif}([-n,n])$ random variables. Why does $(X_n)$ not converge to $0$ in distribution?
My question is on an exercise from Probability Essentials by Jacod and Protter:
18.6 Let $(X_n)_{n \geq 1}$ be a sequence of real valued random variables with $\mathcal{L}(X_n)$
uniform on $[−n, n]$. ...
- 1,067
1
vote
1
answer
55
views
Verify: Uniformly differentiable functions have continuous derivatives
Let $f$ be uniformly differentiable if for all $\varepsilon > 0$, there exists $\delta > 0$ such that for all $x, y \in \mathbb R$,
$$\left \lvert \frac {f(x)-f(y)}{x-y} - f'(y) \right \rvert &...
- 2,161
2
votes
3
answers
104
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 \...
1
vote
2
answers
48
views
Does $\limsup\limits_{x\to\infty}f'(x)=\infty$ imply the nonuniform continuity of $f$ on $(0,\infty)$?
Suppose $f$ is differentiable on $(0,\infty)$ and $\limsup\limits_{x\to\infty}f'(x)=\infty$.
Does this imply the function $f$ is nonumiform continuous on $(0,\infty)$?
I know and can prove the ...
- 6,100
0
votes
1
answer
21
views
Show continuity and boundedness for function (and try to simplify it)
Consider the function $f\colon (0,1]\to\mathbb{R}$ given by
$$
f(x):=\begin{cases}n(n+1)x-n, & \textrm{if }x\in\left[\frac{1}{n+1},\frac{1}{n}\right], n\in\mathbb{N}\textrm{ odd}\\-n(n+1)x+n+1, &...
- 385
-1
votes
1
answer
66
views
Show if $f(x) = \ln x$ is uniformly continuous on the interval $(0, 1)$ [closed]
Show if $f(x) = \ln x$ is uniformly continuous on the interval $(0, 1)$.
How I have started:
Let $\epsilon>0$. There exits $\delta>0$ such that $x,y\in(0,1)$ with $|x-y|<\delta$ $\implies|f(...
- 15
1
vote
1
answer
96
views
If $(f_n)$ converges pointwise to $f$, then $f$ is uniformly continuous
I am preparing for my Analysis $2$ final and am looking over some problems on past homeworks and exams that I could not solve. This is one of them:
Prove that if $(f_n)_{n\in\mathbb{N}}$ is ...
- 2,567
0
votes
0
answers
16
views
Difference between uniform continuity and continuity - taking the UC $\delta$ as the min of the C $\delta$ as $x\rightarrow 0$
What is the difference between UC and C ?
Well most of us know that in UC, $\delta$ depends only on $\epsilon$ whereas in Continuity $\delta$ depends on $\epsilon$ and on $x$
So, we have $\delta =f(\...
- 618
0
votes
1
answer
24
views
Showing the integral of a function is uniformly continuous on a non-upper-bounded interval
I am preparing for an exam, and in finding an old exam for my course, I have run into following problem:
We are to show that the function
$$u(x)=\int_0^x \frac{\sin(x-y)}{1+y^2} dy$$
is either not ...
0
votes
1
answer
45
views
Is $f$ (uniformly) continuous on $(0, k]$ if and only if $1/f$ is (uniformly) continuous on $[k, +\infty)$?
For $f: A \subset \mathbb{R} \setminus\{0\} \to \mathbb{R} \setminus\{0\}$, is $f$ uniformly continuous on $(0, k]$ if and only if $1/f$ is uniformly continuous on $[k, +\infty)$?
The examples I've ...
- 2,161
0
votes
1
answer
46
views
Show that if $f\in E$, then $f$ is uniformly continuous.
Here's the problem statement.
Let $E$ be the set of functions $f:[0,1]\to \mathbb{R}$ such that $K(f)=\text{sup}${$\frac{|f(x)-f(y)|}{|x-y|}: x,y\in [0,1] \text{ with } x\neq y$}$<\infty$.
(i) ...
user1086223
0
votes
2
answers
79
views
Understanding proof of theorem 1, appendix to chapter 8 in Spivak's Calculus
I am trying to understand Spivak's proof of theorem 1 in the appendix to chapter 8. He uses a lemma in the proof, which in turn uses two statements (i) and (ii).
Prerequisites
Suppose that we have ...
- 329
1
vote
1
answer
48
views
function with summable derivative is uniformly continuous? [closed]
If we have the function $f: I \to \mathbb{R}$ with derivative $f’: I \to \mathbb{R}$ such that: $I \subseteq \mathbb{R}$ $f’ \in L^{1}(I)$
can we deduce that $f$ is uniformly continuous?
Of ...
- 21
2
votes
0
answers
55
views
Let $(X, d)$ be a separable metric space. Then the space of all real-valued bounded uniformly continuous functions on $X$ is separable
It's well-known that
Theorem 1 A metric space is compact IFF its space of bounded, continuous, real-valued functions is separable in the uniform topology.
I would like to prove its complementary ...
- 3,914
1
vote
0
answers
27
views
0 as point of non-uniform convergence for $f_n(x)=nxe^{-nx^2}$
I was solving a question on uniform convergence of sequences and series of functions and came across the following question:
Show that if $f_n(x)=nxe^{-nx^2}$, the sequence ${f_n}$ is not uniformly ...
- 73
0
votes
1
answer
20
views
Showing that this limit converges in an "uniform sense"
In the middle of a proof I came across the following statement that I was not able to prove.
Let $\epsilon>0$, and consider the function $f(x,t) = |e^{-cx^2t}-1|$ for $x\in\mathbb{R},t>0$, ...
- 137
1
vote
1
answer
81
views
If $f_n \to f$ and for every $n > 0$ $f_n$ is uniformly continuous then f is uniformly continuous. Where did my proof fail?
Let $[a,b] \subseteq \Bbb R$, let $f_n:[a,b] \to \Bbb R \space\space$ such that $\forall n>0 \space\space f_n$ is uniformly continuous on $[a,b]$. Let $f:[a,b] \to \Bbb R$, Suppose that $f_n \to f \...
- 113
0
votes
1
answer
80
views
Metric spaces: patching uniform continuity from subsets
Let $(X, d)$ be a metric space, and $A$, $B$ be closed subsets of $X$ (not necessarily disjoint) with $A \cup B = X$.
Suppose I have a function $f : X \to Y$, for $(Y, d')$ some other metric space, ...
- 8,899
1
vote
1
answer
37
views
Uniform continuity and seminorms
I recall that a seminorm is basically a norm that is not necessarily positive definite.
Let $(E,(p_n)_{n\in\mathbb{N}})$ be a Fréchet space, meaning each $p_n$ is a seminorm and if we equip $E$ with ...
- 133
1
vote
0
answers
33
views
proof: show that f'(x) is uniformly continuous on I.
Question:
Let I = (a, b) be a nonempty open interval, f is differentiable on I. for each ε > 0, there exists δ > 0 such that for all x, y ∈ (a, b) satisfying 0 < |x − y| < δ we have
$\vert{...
- 11
0
votes
0
answers
25
views
Uniform continuous, equicontinuity, and Banach-Steinhaus
Banach-Steinhaus is a sufficient condition for a family of operators to be equicontinuous.
Here, in Does uniform continuity on a compact subset imply equicontinuity?
We have a function $f_n(x) = x^n$ ...
- 95
1
vote
0
answers
46
views
Riemann Integral and Limits in Exercise 5 Section 1B of Measure, Integration & Real Analysis by Sheldon Axler
I am fairly new to real analysis and shyly looked into the book specified in the title because I loved Linear Algebra Done Right also by Axler.
Now I think Heine-Borel makes $[0,1]$ compact and a ...
- 11
2
votes
1
answer
47
views
Existence of continuous functions with finitely many prescribed values
This question seems very basic but I have no clue how to show this statement nor have I been able to find some references for it.
Let $X$ and $Y$ be two uniform Hausdorff spaces (i.e. completely ...
- 43
0
votes
0
answers
62
views
On Bartle's 'Elements of Integration' Exercise 7.S
As I haven't found a reference solution for this exercise on the net and I am not sure whether my argumentation is sound, I decided to post my solution here.
My question is twofold:
Are there errors ...
0
votes
0
answers
66
views
Is there a topology on the reals such that the continuous functions of that topology are precisely the uniformly continuous functions?
This is a follow-up to my previous question, here: Is there a topology on the reals such that the continuous functions of that topology are precisely the differentiable functions?. Now, I am asking ...
- 16.8k
-1
votes
1
answer
67
views
What is an example of a function that is uniformly continuous but not continuous on an interval $[a,b]$? [closed]
In the Appendix to Chapter 8 of Spivak's Calculus, entitled "Uniform Continuity", there is the following theorem
If $f$ is continuous on $[a,b]$, then $f$ is uniformly continuous on
$[a,b]$
...
- 4,134
0
votes
0
answers
62
views
Undergraduate Real Analysis problem books that do not require Topology
I have seen a lot of suggestions of real analysis textbooks on StackExchange. But unfortunately, because my university course (I am in undergraduate year 2) does not teach Topology before Analysis, ...
- 141
0
votes
0
answers
43
views
Not Uniformly continuous function of two variables
I am leading with the following function, defined by:
$$f(x,y)=\frac{x^2y^2}{x^2y^2-(x^2+y^2)},f(0,0)=0$$
I have showed that it is continuous but now I was asked if it is uniformly continuous in the ...
- 359