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".
2,172 questions
0 votes
2 answers
48 views
Uniformly continuous in interior can be countinously extened [closed]
Problem is: The function $f(z)$ is uniformly continuous in the disk $D=\{|z|<1\}$. Prove that for any point $\zeta$ such that $|\zeta|=1$ and any sequence $z_n\to \zeta$, $z_n \in D$ there exists ...
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1 vote
1 answer
90 views
Uniform Continuity Real Analysis Proof
Let $a < b < c < d$ be real numbers, and suppose $f : (a, d) \to \Bbb{R}$ is a function such that $f$ is uniformly continuous on $(a, c)$ and also uniformly continuous on $(b, d)$. Prove that ...
0 votes
0 answers
88 views
How to construct a counter example of the question?
Problem:If $f(x)\in C(0,1]$,and for every certain and fixed value $x_0\in(0,1]$,the limit $\lim_{n \to \infty}f(\frac{x_0}{2^n}) $ exists. Can we get the limit $\lim_{x \to 0^+}f(x)$ exists? And I ...
- 161
4 votes
1 answer
170 views
Properties of continuous functions on $(0,\infty)$ with $f(x) = f(2x)$ for all $x$.
The problem: Let $\mathscr{S}$ be the family of continuous real valued functions on $(0,\infty)$ defined by: $$ \mathscr{S} := \left\lbrace f : (0,\infty)\to\mathbb{R} \,\middle|\, f(x) = f(2x), \...
- 342
2 votes
1 answer
43 views
$\dot H^1$ bound for error Riemann sums
$\def\d{\mathrm d} \newcommand\abs[1]{\left| #1 \right|}$ Consider $f \in C^1 ([0, 1], \mathbb R)$, and a uniform partition of $[0, 1]$ : $x_n := \dfrac nN$, $0 \le n \le N$. The usual bound on the ...
- 328
2 votes
1 answer
271 views
Can a continuous nowhere differentiable function be uniformly continuous and unbounded?
I came up with the following problems myself (a) Give an example of a continuous nowhere differentiable function $f:\mathbb{R}\to\mathbb{R}$ that is bounded and not uniformly continuous. (b) Give an ...
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0 votes
0 answers
31 views
An embedding of uniformly local Sobolev spaces
Given a Hilbert triplet $V\hookrightarrow H\hookrightarrow V'$, we consider the space $H^1_{uloc}([0,\infty);V')$ as the space of measurable functions $f:[0,\infty)\to V'$ such that $$sup_{t\geq0}\...
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0 votes
0 answers
19 views
What Is a UV map to a higher dimensional space that enables tiling continuity of 3D cubes?
This is expanding on the Idea of This Blender Stack Exchange post Describing tiling 2D noise using a projection into a higher-dimensional space. As a continuation of this method, would it be possible ...
- 117
5 votes
1 answer
370 views
Example of Oscillating Faster and Faster and No Infinite Limit but Uniform Continuous
I have seen that the continuous functions $$ \sin(x^2) \quad\text{and}\quad x\sin{x} $$ are not uniformly continuous on $[0,\infty)$. Their non uniform continuity is easy to prove with sequences, but ...
- 329
2 votes
0 answers
54 views
Initial uniformity induced by continuous functions on $\mathbb R$?
Background: It is known from the theory of uniformizable spaces that any topological space $E$ can be associated to a uniform structure, by considering the family of pseudometrics $\{d_f\}_{f\in C}$, ...
- 123
0 votes
1 answer
56 views
Problem following proof of Heine(-Cantor)'s theorem
The theorem (in this form) states Let $A \in \mathbb{R}$ be a compact set and $f: A \to \mathbb{R}$ continuous. Then $f$ is uniformly continuous. Using this equivalence: $[f$ is not uniformly ...
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1 vote
1 answer
141 views
Functional Equations $f\left( x \right) =f\left( \frac{x^{2}}{2} \right)$. [duplicate]
Let $f \in C(2, +\infty)$ and satisfy $f(x) = f\left( \frac{x^2}{2} \right)$ for all $x \in (2, +\infty)$. Determine whether $f$ is uniformly continuous on $(2, +\infty)$. My idea Consider $$f\left( x ...
user1156805
0 votes
1 answer
81 views
Continuation of a function to the boundary
Let we have a function $f(x,y)\in C^1(K),$ where $K=[0;1]\times[0;1)$ (square without upper segment) and let's say we have a continuous partial derivative in the entire square $\displaystyle\frac{\...
- 169
0 votes
1 answer
95 views
Why is $\lim_{h \rightarrow 0} |f(t+h) - f(t)| = 0$ sufficient for proving uniform continuity?
The second page of https://web.ma.utexas.edu/users/gordanz/notes/oldchar.pdf proves that the characteristic function $\varphi$ of a random variable is uniformly continuous. The proof is that $$|\...
- 969
14 votes
2 answers
2k views
Does there exist a computer program that is able to determine whether a given function is uniformly continuous?
I was wondering if someone might know whether there exists (or whether it is even possible) for a computer program to process a given function and determine whether it is uniformly continuous. I have ...
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