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".
301
questions with no upvoted or accepted answers
9
votes
1answer
158 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},\...
8
votes
0answers
492 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
0answers
52 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
0answers
76 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
$$
d_Y(f(x),f(y)) \le ...
5
votes
0answers
257 views
Baby Rudin Exercise 4.13 Alternate Proof Verification
I would like to know if my proof of ex 4.13 is correct. Thanks!
Exercise 4.13 in Rudin asks: Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function ...
5
votes
1answer
917 views
Does uniform continuity of the differential imply uniform differentiability?
Let $E \subseteq \mathbb{R}^n$ be an open subset. $f:E \to \mathbb{R}$ be differentiable, and suppose that $\nabla f$ is uniformly continuous.
Is it true that $f$ is "uniformly differentiable"? i....
4
votes
0answers
34 views
Groups where every continuous function is uniformly continuous
Let $G$ be a topological group, and $(X, d)$ a metric space. The function $f : G \to X$ is left uniformly continuous if for all $\varepsilon > 0$ there exists an identity neighbourhood $U$ such ...
4
votes
1answer
56 views
I am using neighborhood balls to define continuity. Are these definitions of pointwise continuous and uniform continuous correct?
I seem to understand topology more than analysis and was wondering if these definitions of continuity, which to me have more a topological flavor, are correct.
Suppose $X$ and $Y$ are metric spaces ...
4
votes
0answers
52 views
Uniform continuity of $\frac{1}{2} \sin{ \frac{x}{2}}- \frac{1}{x}$
Let $f(x)=\frac{1}{2} \sin{ \frac{x}{2}}- \frac{1}{x}$ if $0<x \leqslant \pi$ and $f(x)=0$ if $x=0$.
I took the sequences $$x_n=\frac{1}{n}$$ $$y_n=\frac{1}{n^2}$$
We have that $x_n-y_n \...
4
votes
0answers
312 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
1answer
610 views
If $f$ is continuous on $[a,b]$ then $f$ is uniformly continuous on $[a,b]$.
So I want to prove that continuity on $[a,b]$ implies uniform continuity with only using the least upper bound property of the reals. I know the basic idea of this, but am getting confused with ...
4
votes
1answer
126 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$, ...
4
votes
0answers
91 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
1answer
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 ...
3
votes
0answers
55 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 ...
3
votes
1answer
77 views
How to check whether $\sin (x \sin x)$ is uniformly continuous or not on $\mathbb R$.
How to check whether $\sin (x \sin x)$ is uniformly continuous or not on $\mathbb R$.
My Try: I took two sequence namely $t_n = \frac{n\pi}{2} + \frac {\pi}{n}$ and $z_n = \frac{n\pi}{2}$ . Now $|...
3
votes
1answer
348 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 ...
3
votes
0answers
61 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,$$...
3
votes
0answers
762 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 ...
3
votes
0answers
76 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 $[a, b]$...
3
votes
0answers
44 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 ...
3
votes
0answers
446 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 $...
2
votes
0answers
41 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 ...
2
votes
0answers
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=...
2
votes
0answers
29 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 ...
2
votes
0answers
23 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 ...
2
votes
0answers
47 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 = \...
2
votes
1answer
37 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 ...
2
votes
0answers
42 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\...
2
votes
0answers
31 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 $[...
2
votes
0answers
96 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
votes
0answers
47 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!
...
2
votes
0answers
38 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 ...
2
votes
0answers
41 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 ) ...
2
votes
0answers
87 views
About continuity on $\mathbb Q\cap [0,1]$
Decide if the following statements are true or false and justify yor answer.
(a) Every bounded function $f:\mathbb Q\cap [0,1]\to\mathbb R$ is continuous.
(b) Every continuous function $g:\mathbb Q\...
2
votes
0answers
466 views
Prob. 13, Chap. 4 in Baby Rudin: Extension of a uniformly continuous real function from a dense subset to the entire space
Here is Prob. 13, Chap. 4 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real ...
2
votes
0answers
131 views
Is trace map uniformly continuous?
We have trace map as $tr : M_n(F) \to F.$
Then using trace map is linear map I get $||tr(A)-tr(B)||=||tr(A-B)|| \le ||tr||||A-B||.$
Since I know that trace is a continuous map, I have $||tr||$ as ...
2
votes
1answer
155 views
Proof of the “second half of the Heine-Cantor theorem”
A pseudometric on a set $X$ is a metric except it need not be Hausdorff. A gauge $\mathcal D_X$ on $X$ is a an ideal of pseudometrics, i.e.:
$\mathcal D_X \neq \emptyset$
$d_1,d_2 \in \mathcal D_X \...
2
votes
0answers
237 views
Metric Spaces: a Uniformly Continuous Function Preserves Total Boundedness.
I've seen references to this statement but not a proof. I've worked up a proof myself and would appreciate confirmation or correction.....
Let $f: (X, d_x) \to (Y, d_y)$ be uniformly continuous on $X$...
2
votes
0answers
176 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 ...
2
votes
0answers
161 views
Uniform Convergence of a Function Related to the Characteristic Function of a Random Variable
This question is related to a previous question I asked:
Using the Uniform Continuity of the Characteristic Function to Show it's Differentiable
Suppose we have a characteristic function $\varphi$...
2
votes
0answers
63 views
Let $(f_n)$ be a sequence of functions on $[a,b]$ such that each $f_n$ is continuous on $[a,b]$ and differentiable on $(a,b)$
Let $(f_n)$ be a sequence of functions on $[a,b]$ such that each $f_n$ is continuous on $[a,b]$ and differentiable on $(a,b)$ and there exists $M>0$ such that $|f_n'(x)|ā¤M$ for all $x \in (a,b)$ ...
2
votes
0answers
127 views
Uniform Continuity: How can I show a continuous function $f: ]0,1] \rightarrow R $ is uniformly continuous if $\lim_{x \, \searrow \, 0}f(x) $ exists?
Let $f: ]0,1] \rightarrow R $ be a continuous function. How can I show that $f$ is uniformly continuous exactly then, when $\lim_{x \, \searrow \, 0}f(x) $ exists? I understand this requires a two ...
2
votes
0answers
80 views
Uniform continuity, a direct proof
A common question in analysis is: Let $I$ be an open interval in $\mathbb{R}$. Show that a continuous function $f:I \to \mathbb{R}$ is uniformly continuous iff for all sequences $(x_n), (y_n) \subset ...
2
votes
0answers
49 views
How to check whether $f$ is uniformly continuous
Suppose that $f:\Bbb R\to \Bbb R$ is a function satisfying $|f(x)-f(y)|\le |x-y|^\beta $.
If $\beta >0$ show that $f$ is uniformly continuous.
Attempt:
Let $\epsilon>0$ be given. Then choose $...
2
votes
0answers
827 views
Why do uniformly continuous functions form a Banach space (with the sup norm)?
Let $A$ be a metric space, and $U_{b}(A)$ the set of all bounded and uniformly continuous functions $g:A\rightarrow \mathbb{F}$.
Let $||\cdot||$ be the sup norm with $||g||=\sup_{x\in A}|g(x)|$.
...
2
votes
0answers
176 views
$f$ is uniformly continuous only if $g$ is constant
Let $g:\mathbb R\to\mathbb R$ be continuous and define $f:\mathbb R^2\to\mathbb R$ by $f(x_1,x_2)=g(x_1x_2)$. Show that $f$ is uniformly continuous only if $g$ is a constant function.
I'm not sure ...
2
votes
0answers
117 views
Equivalent definition of continuity, unif.continuity, equicontinuity, etc
I am trying to summarize some definitions regarding the different types of continuity I know, in my own words, and I would like to know if you think they are correct (that is, if they are equivalent ...
2
votes
0answers
303 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 ...
2
votes
0answers
99 views
Condition for term-by-term differentiation of a non-convergent series
In a problem I have seen, a series $\sum_n u_n(x)$ with $$ f_n(x) = \frac {\log(1+n^4x^2)}{2n^2}$$
here $\sum u_n'(x)$ is not uniformly convergent, BUT
If $f '(x) = \lim_{n\to\infty} f_n'(x) $
...
