My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Thanks Tree1Thanks
  • 1 Post By romsek
Reply
 
LinkBack Thread Tools Display Modes
March 11th, 2018, 09:24 PM   #1
Senior Member
 
Joined: Nov 2015
From: hyderabad

Posts: 232
Thanks: 2

Question on Sets

If $f(x) = g(x)$ $\forall x \epsilon Q$ then $f(x) = g(x)$ $\forall x \epsilon R$
Is this always true ?
Since the set of Reals contains both Rational & Irrational numbers, I feel it is not always true. But someone kindly help me with this problem.

Thank you!
Lalitha183 is offline  
 
March 11th, 2018, 10:55 PM   #2
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 2,011
Thanks: 1044

It's rather trivial to construct a functions $f(x),~g(x)$ where it's not true

$f(x) = \begin{cases}1 &x \in \mathbb{Q} \\0 &x \not \in \mathbb{Q} \end{cases}$

$g(x) = 1,~x \in \mathbb{R}$

$x \in \mathbb{Q} \Rightarrow f(x) = g(x) = 1$

$x \not \in \mathbb{Q} \Rightarrow f(x) = 0 \neq 1 = g(x)$
Thanks from Lalitha183
romsek is offline  
March 12th, 2018, 05:30 AM   #3
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 398
Thanks: 212

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
Originally Posted by Lalitha183 View Post
If $f(x) = g(x)$ $\forall x \epsilon Q$ then $f(x) = g(x)$ $\forall x \epsilon R$
Is this always true ?
Since the set of Reals contains both Rational & Irrational numbers, I feel it is not always true. But someone kindly help me with this problem.

Thank you!
It is definitely not true as was pointed out. However, it is interesting that the only hypothesis which needs to be added to make it true is continuity. If $f,g$ are continuous functions defined on $\mathbb{R}$ which agree on the rationals, then they must agree on all of $\mathbb{R}$.
SDK is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
question, sets



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Infinite sets question shunya Elementary Math 2 October 29th, 2015 08:47 AM
Quick question on Sets? maths4computing Applied Math 3 August 25th, 2013 12:30 PM
Help! Question regarding open sets chappyform Real Analysis 2 April 28th, 2013 05:19 PM
I need help with these question on Sets... Britmouth Applied Math 6 August 31st, 2010 06:08 PM
sets theroy question sepehr Applied Math 3 April 3rd, 2010 02:46 AM





Copyright © 2018 My Math Forum. All rights reserved.