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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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}, \: \...
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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
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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
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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
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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
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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
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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
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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
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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
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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 $\varphi$...
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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 $|f_n'(x)|≤M$ for all $x \in (a,b)$ ...
2
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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
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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
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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
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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
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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 ...
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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) $ ...

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