For questions involving the concept of uniform continuity, that is, "the $\delta$ in the definition is independent of the considered point.

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12 views

Suppose $f$ is differentiable and not uniformly continuous show that $|f'|$ is not bounded by $R$

Show for any constant $C$, $|f'(t)|> C$ for some $t$ exits in $R$. I can see why this is true but just cannot write up a good solution.
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7 views

Normal transitional pdf Wiener process continuity mistake and what other standard pdfs are used

I need you to tell me where I am making a mistake in the following: $$f_{1|1}(x_2,t+\Delta t|x_1,t) = \frac{1}{\sqrt{2\pi\Delta t}}e^-{\frac{(x_2-x_1)^2}{2\Delta t}}$$ If I let $\Delta x = x_2-x_1$, ...
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0answers
23 views

continuity of a piece wise function defined partially on a closed interval

using epsilon delta definition prove that $f(x)=\left \{ \begin{array}{cc} 2 & : x \in[0,1]\\ 1 & : x=-1 \end{array}\right.$ is continuous on $E= [0,1] \cup \{-1\}$. Here is my attempt. I ...
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1answer
43 views

Determine continuity and differentiability of the real function $f(x)=\sum\limits_{n\geq1}\frac{1}{n^x}$

I've been asked to analyze the domain, continuity and differentiability of the function $$f(x)=\sum\limits_{n\geq1}\frac{1}{n^x}$$ I've already shown that this function is defined for every real ...
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1answer
50 views

Uniform continuity with integral being finite

Let $f$ be a real valued uniformly continuous function on $\mathbb{R}$ that is lebesgue integrable. Show that $\lim_{|x|\rightarrow \infty}f(x)=0$. Suppose that ...
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1answer
21 views

Polynomials and Lipschitz function

Let $f(x) = x^4 + 11x^2 + 9x -5$ and let $M > 0$. Show that f is a Lipschitz function on the interval $[-M, M]$ I honestly cannot figure out how to start this proof. Nothing similiar is in the ...
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0answers
9 views

Nonuniform Continuity Criteria with cos(1/x)

Use the nonuniform continuity criteron to show that $f(x) = cos\left(\frac{1}{x}\right)$ is not uniformly continuous on $(0, \infty)$ My proof: Let $(x_n)$ be defined by $x_n = \frac{1}{2n\pi}$ ...
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2answers
15 views

Uniformly Continuous functions and paths

Consider a function $ \gamma(t) = (f_1(t), f_2(t), ... , f_k(t))$ for $t \in [a,b]$. So, $f_i$ are real-valued functions on $[a,b]$. Then $\gamma$ is continuous, i.e. a path, if and only if each ...
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0answers
37 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 ...
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1answer
23 views

(Uniform) continuity of push-forward operator

I am wondering about extension of the the answer given here. Namely, suppose $U$, $V$ are Polish spaces and $F:U→V$ is uniformly continuous. Does this mean that the push-forward operator $F_*: ...
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2answers
92 views

Is the function $f:\mathbb C \to \mathbb C$ defined as $f(z)=\sqrt z$ uniformly continuous ?

Is the function $f:\mathbb C \to \mathbb C$ defined as $f(z)=\sqrt z$ uniformly continuous ? Is it uniformly continuous on some non-compact restriction of the complex plane (other than $(0,\infty)$ ) ...
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0answers
52 views

Give an example of the following or explain why it's impossible (Real Analysis):

A sequence $(f_n)$ of differentiable functions such that both $(f_n)$ and $(f'_n)$ converge uniformly but $f = \lim f_n$ is not differentiable at some point. Thoughts so far: We are taking the ...
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1answer
33 views

Least involved proof for continuos functions => uniform continuos functions on [a,b]

I have been looking at this proof in my textbook and seem to always get lost in its logic, its roughly 3 pages long. The proof is: If f is continuous on a closed interval [a,b], then f is uniformly ...
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1answer
36 views

Bolzano-Weierstrass and Uniform Continuity

I don't really know where to begin here. Any help is appreciated, thank you. "Let K be a compact set in $\mathbb R^n$. You may assume the Bolzano-Weierstrass theorem: every sequence in K has a ...
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2answers
27 views

Uniformly continuous independent of metrics?

Let $(X,d)$ and $(Y,e)$ be metric spaces. A map $f:X\to Y$ is uniformly continuous if for each $\epsilon>0$ there exists $\delta >0$ such that whenever $d(x,y)<\delta$ we have ...
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0answers
19 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, ...
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2answers
43 views

If $f_n\to f$ uniformly and $f_n$ are differentiable, is $f$ differentiable?

Suppose $f_n$ converges uniformly to $f$ and $f_n$ are differentiable. Is it true that f will be differentiable? My initial guess is no because $f_n= \frac{\sin(nx)}{\sqrt n}.$ Is this right? And ...
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2answers
35 views

Prove that if $f$ and $g$ are uniformly continuous on A and are both bounded on A, then $fg$ is uniformly continuous on A.

Let $f$ and $g$ be uniformly continuous on A. Then given $\epsilon >0$ there exists a $\delta_{1} > 0$ such that if $|x-y| < \delta_{1}, \forall x,y \in A$, then $|f(x)-f(y)| < ...
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2answers
57 views

How to prove that continuous function do not necessarily preserve cauchy sequences

I am trying to construct a proof that continuous function do not preserve Cauchy sequences Every proof I can find is disprove by counter example, which is great but these counter examples cannot be ...
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2answers
70 views

Problem about uniform continunity on $[0,\infty)$

The question is the following: $f(x)$ is uniformly continuous on $[0, \infty)$ and for any $x > 0$, $\lim\limits_{n\to \infty}f(x+n) = 0$, where $n \in \mathbb{Z}_{>0}$. Prove that ...
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0answers
39 views

Proof that if $f_n \rightarrow f$ uniformly on $D$, and each $f_n$ is uniformly continuous on $D$, then $f$ is uniformly continuous on $D$.

I came across the following statement - that if $f_n \rightarrow f$ uniformly on $D$, and each $f_n$ is uniformly continuous on $D$, then $f$ is uniformly continuous on $D$. How could I prove this ...
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1answer
61 views

$\forall \epsilon >0 \exists$ a piecewise linear function $g$% such that for a continous function $f$

How would I show the following Edit A function is called piecewise linear if it is (1)Continuous (2)Its graph consists of finitely many linear segments Prove that a continuous function on an ...
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2answers
37 views

continuous and uniformly continuous

I remember proving $x^3$ is not uniformly continuous on $\mathbb{R}$. Then I read the proof of a theorem: Suppose $D$ is compact. Function $f: D \rightarrow \mathbb{R}$ is continuous on $D$ if and ...
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1answer
20 views

Uniform continuity piece wise lienar

A function is called piecewise linear if it is (1)Continuous (2)Its graph consists of finitely many linear segments Prove that a continuous function on an interval [a,b] is the uniform limit of a ...
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2answers
54 views

Is there an uniformly continuous $[0,\infty) \rightarrow \mathbb R$ surjection?

I think there isn't but I can't write a proof. I tried assuming that such function is (as it must be) continuous, and showing that it can't be uniformly cont., but I'm not sure what to do.
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1answer
32 views

Let $f: X \subset \mathbb{R}^n\to \mathbb{R}^m$. Then, $f$ is uniformly continuous if, and only if, for every sequence…

Let $f: X \subset \mathbb{R}^n\to $. Then, $f$ is uniformly continuous if, and only if, for every sequence, $(x_n)_{n\in\mathbb{N}}$, $(y_n)_{n\in\mathbb{N}}$ such that $d(x_n,y_n) \to 0$, then ...
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1answer
52 views

Asymptotics of uniformly continuous functions over $\mathbb R$

For $\alpha >1$, $x^\alpha$ is not uniformly continous over $\mathbb R$ , essentially because $\forall \delta>0, \lim_n (n+\delta)^{\alpha}-n^{\alpha}=\infty$ I don't have a clear intuition ...
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1answer
29 views

uniformly continuous functions on infinite open interval $(0, \infty)$

Is $\sin x/x$ uniformly continuous on $(0, \infty)$? Can any one help me proving this? $d/dx(\sin x/x)$ is not bounded on $(0, \infty)$. How to proceed then?
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2answers
32 views

What is the difference between $\implies$ and $\land$ in definition of continuity

Can someone please explain to me what would happen if instead: Given $D \subset \mathbb{R}$ $\forall \epsilon > 0, \exists \delta$ such that $\forall x, x_o \in D, |x-x_o| < \delta \implies ...
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1answer
41 views

Showing $\tan x$ is uniformly continuous in $[0,\pi/4]$

Since any function which is continuous on a closed and bounded interval is uniformly continuous, in that way the given function is uniformly continuous. And in the $\epsilon, \delta$ method I tried ...
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1answer
35 views

Mean-Value Theorem Hypothesis

I know the mean value theorem hypotheses are: If $f: [a,b] \mapsto R$ is continuous on $[a,b]$ and differentiable on $(a,b)$. But can I still invoke MVT under the hypothesis that: $f: (a,b) \mapsto ...
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1answer
43 views

Explain why the following conjecture for $f(x)=[x]+(x-[x])^{[x]}$ is not correct?

Explain why the following conjectures for $f(x)=[x]+(x-[x])^{[x]}$ are not correct ($x\in(0,\infty)$) Conjecture a: $\lim_{x\rightarrow\infty}(f(x)-[x])=0.$ Conjecture b: $f$ is uniformly ...
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1answer
31 views

On differentiable functions on real line satisfying $f'(x)\ge f(x)^2 , \forall x>0$

Does there exist a real valued differentiable function $f$ on real line such that $f'(x) \ge f(x)^2 , \forall x >0$ ? If such a function exist , must it be twice differentiable or at least ...
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1answer
34 views

Given a continuous function with an asymptote, prove that the function is uniformly continuous.

I state the exercise: Given $f: [0, + \infty) \rightarrow R$, f continuous. Prove that if $\lim_{x \rightarrow \infty} f(x) = \lambda$, where $\lambda \in R$ then $f$ is uniformly continuous. My ...
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3answers
242 views

Counterexample: Continuous, but not uniformly continuous functions do not preserve Cauchy Sequences

I want to prove this: There exists a continuous function $f:\mathbb{Q}\to\mathbb{Q}$, but not uniformly continuous, and a Cauchy sequence $\{x_n\}_{n\in\mathbb{N}}$ of rational numbers such that ...
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3answers
75 views

If a sequence $(f_n)$ do not converge uniformly, then no subsequence of $(f_n)$ converge uniformly

Can someone verify this claim: If a sequence $(f_n)$ do not converge uniformly, then no subsequence of $(f_n)$ converge uniformly I saw this in a proof, where $f_n(x) = x^n$ on $x \in [0,1]$, ...
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3answers
76 views

How is it not the case that every continuous function is uniformly continuous?

What's wrong with my following proof? Suppose $f:\Bbb R\to \Bbb R$ is continuous. Take $x,y\in \Bbb R$. Given $\epsilon >0$ there exists $\delta_1,\delta_2$ such that $$ 0<|x-x_0|<\delta_1 ...
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2answers
33 views

What's the difference between uniformly equicontinuous and uniformly continuous?

I am very confused. Thanks in advance. Our definition is that: Uniformly Equicontinuous: $\forall \epsilon>0,\exists\delta>0 \ such \ that \ |s-t|< \delta \ and \ n \in \mathbb{N} \ then \ ...
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1answer
53 views

A sequence of continuous functions, $\{f_n\}$ converges pointwise to a discontinuous function $g$, then $\{f_n\}$ does not converges uniformly to $g$?

If a $\{f_n\}$ a sequence of continuous functions, converges pointwise to a discontinuous function $g$, does it imply that $\{f_n\}$ does not converges uniformly to $g$?
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28 views

What if we remove some conditions in this statement, will it still be true?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$. We can show that that if $f:X\to Y$ is uniformly continuous, then $f$ maps Cauchy sequences to Cauchy sequences, i.e., $$(x_n)\text{ is ...
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1answer
33 views

How to show Cantor function is uniformly continuous?

Cantor function $f$ is defined by $f: \Delta \to [0,1]$ by $$f(\sum_{n=1}^{\infty} \frac{2b_n}{3^n}) = \sum_{n=1}^{\infty} \frac{b_n}{2^n}, b_n \in \{0,1\}$$, where $\Delta$ is a Cantor set ...
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1answer
32 views

T/F: pointwise equicontinuous $\Leftrightarrow$ all $f_n$ continuous, uniform equicontinuous $\Leftrightarrow$ all $f_n$ uniformly continuous

Is it true that: Given a sequence of functions $(f_n)$ on a set not necessarily compact pointwise equicontinuity $\Leftrightarrow$ all $f_n$ are continuous and, uniform ...
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1answer
62 views

Uniformly continuous functions sequence $f_n(x)$ converges uniformly to a uniformly continuous function $f(x)$? [duplicate]

We know that if continuous functions sequence $g_n(x)$ converges uniformly to $g(x)$, then $g(x)$ is continuous function. But what if uniformly continuous functions sequence $f_n(x)$ converges ...
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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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1answer
35 views

$f$ is uniformly continuous on $X$ if and only if $d(x_n,z_n)\to0\implies\rho(f(x_n),f(z_n))\to0.$

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$. Prove that $f$ is uniformly continuous on $X$ if and only if for any sequences $(x_n)$ and $(z_n)$ in $X$, ...
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1answer
37 views

How can I prove this function is uniformly continuous?

The function is $$ f : B((0,0),1) \rightarrow \Bbb R,\,f(x,y)={xy\over{x-1}}. $$ I tried with sequences: Let be $ \{(x_k,y_k)\} $ and $ \{(x'_k,y'_k)\} $ sequences of $B((0,0),1)$ such that $ ...
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1answer
40 views

$f$ is uniformly continuous on $[a,b]$ and $[b-1,c]$ $\Rightarrow$ uniformly continuous on $[a,c]$

$f$ is uniformly continuous on $[a,b]$ and $[b-1,c]$ $\Rightarrow$ uniformly continuous on $[a,c]$ My thoughts: Without loss of generality we only need to show that it's uniform if $x_1 \in ...
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1answer
31 views

Uniform continuity of $\sin(x+y)$

The function is: $f(x,y)=\sin(x+y)$. Is it uniformly continuous in $\mathbb{R}^2$? How do I prove it?
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1answer
38 views

From continuity to uniform continuity

Consider a convex, open, non-empty set $B \subset \mathbb{R}^k$ and a function $V: B\rightarrow \mathbb{R}$ convex (and, hence, continuous) in $B$. Consider $\epsilon \in \mathbb{R}$, $\epsilon>0$. ...
0
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0answers
27 views

Uniform continuity preserving measure zero

Suppose $f$ is uniformly continuous on $[a,b]$ and $E\subset [a,b]$ such that $\mu(E)=0$ (lebesgue measure) then prove that $f(E)$ is of measure zero. What i have tried so far is : As $f$ is ...