My Math Forum Functions of many variables

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 May 12th, 2013, 04:41 AM #1 Newbie   Joined: Apr 2013 Posts: 11 Thanks: 0 Functions of many variables 1. Find all limit points of sets in $\mathbb{R}^d$: $a) \{ x\in \mathbb{R}^2\ :\ x_1, x_2 \in \mathbb{Q}\}$ $b) \{ x\in \mathbb{R}^2\ :\ x_1\in \mathbb{Z}\}$ $c) \{ x\in\mathbb{R}^3\ :\ x_1^2+x_2^2+x_3^2<1\}$ $d)\{ x\in \mathbb{R}^3\ :\ x_1^2+x_2^2+x_3^2>1\}$ $e) \{ \left( \frac{1}{n},\ \sqrt[n]{n},\ \sqrt[n]{n!}\right)\in \mathbb{R}^3\ :\ n\in \mathbb{N}\}$ 2. Check the existance (and calculate if possible) both limits of iterated functions $f: \mathbb{R}^2\to \mathbb{R}$ in point $(0, 0)$ for: $a) f(x)=\begin{cases}\frac{x_1x_2}{x_1^2+x_2^2} \mbox{ for } x\neq (0, 0) \\ 0 \mbox{ for } x= (0,0)\end{cases}$ $b)f(x)=x_1\cdot \mathbb{D} (x_2)$, where $\mathbb{D}$ is a Dirichlet function. 3. Judge if function $f:\mathbb{R}^d \to \mathbb{R}$ $f(x)=\begin{cases}w(x) \mbox{ for } x \neq 0 \\ 0 \mbox{ for } x = 0\end{cases}$ for $a)\ w(x)=\frac{x_1^2x_2^2}{x_1^2x_2^2+(x_1-x_2)^2},\ d = 2,$ $b)\ w(x)= \frac{x_1x_3+x_2x_3}{x_1^2+x_2^2+x_3^2},\ d=3,$ $c)\ w(x)=\frac{x_1^2x_2}{x_1^2x_2^2+x_1^2+x_2^2},\ d=2,$ $d) \ w(x)=\frac{x_1^2x_2}{x_1^4+x_2^2},\ d = 2,$ is continuous. Thank you.
 May 12th, 2013, 10:13 AM #2 Newbie   Joined: Apr 2013 Posts: 11 Thanks: 0 Re: Some topology sorry about the title, was thinking about something different and made a mistake.
 May 12th, 2013, 10:15 AM #3 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Some topology To what would you like the title to be changed?
 May 12th, 2013, 10:49 AM #4 Newbie   Joined: Apr 2013 Posts: 11 Thanks: 0 Re: Some topology Functions of many variables comes to my mind.
May 12th, 2013, 10:57 AM   #5
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Re: Some topology

Quote:
 Originally Posted by Thy-Duang Functions of many variables comes to my mind.
The title has been changed.

 May 12th, 2013, 11:00 AM #6 Newbie   Joined: Apr 2013 Posts: 11 Thanks: 0 Re: Functions of many variables Thanks a lot.
May 12th, 2013, 03:17 PM   #7
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Re: Functions of many variables

Quote:
 Originally Posted by Thy-Duang 1. Find all limit points of sets in $\mathbb{R}^d$: $a) \{ x\in \mathbb{R}^2\ :\ x_1, x_2 \in \mathbb{Q}\}$
Can you do this one? If not, where are you stuck?

Can you do this problem in one dimension? What are the limit points in $\mathbb{R}$ of

$\{ x\in \mathbb{R}\ :\ x \in \mathbb{Q}\}= \mathbb{Q$

 May 12th, 2013, 11:37 PM #8 Newbie   Joined: Apr 2013 Posts: 11 Thanks: 0 Re: Functions of many variables It's obvious, but don't know how to prove obvious things.
May 12th, 2013, 11:45 PM   #9
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Re: Functions of many variables

Quote:
 Originally Posted by Thy-Duang It's obvious, but don't know how to prove obvious things.
I have the same problem but was never able to articulate it as nicely as you have, so thank you.

Pardon the interuption, i have no idea how to proceed but your questions are fascinating...

 May 13th, 2013, 12:36 AM #10 Newbie   Joined: Apr 2013 Posts: 11 Thanks: 0 Re: Functions of many variables Okay, I've an idea: The sequence $a_n$ that converges to $a$ is a constant sequence if $a\in \mathbb Q ^2$, else $a_n=\lfloor a \rfloor + b_n$ where $b_n$ gives first $n$ digits of $\{a\}$

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