My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
February 6th, 2012, 12:36 PM   #1
Member
 
Joined: Jan 2012

Posts: 63
Thanks: 0

continuous for 1/x

For the function f(x)=1/x, prove that with ? =1, there does not exist a ?>0 s.t. the definition for continuity "for all ? > 0, exist a ? > 0 s.t. |x-c|<? then |f(x)-f(c)|<?" holds for all points c and x in (0,infinity).
frankpupu is offline  
 
February 7th, 2012, 06:00 AM   #2
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,947
Thanks: 1139

Math Focus: Elementary mathematics and beyond
Re: continuouse for 1/x

Quote:
Originally Posted by frankpupu
holds for all points c and x in (0, infinity)
Should be (0, 1), no?
greg1313 is offline  
February 9th, 2012, 05:43 PM   #3
Global Moderator
 
Joined: Dec 2006

Posts: 20,753
Thanks: 2136

No, infinity will do.

If such ? > 0 exists, let c = ?/(2? + 2) and x = c + ?/2 = (? + 2?)/(2? + 2), so that |x - c| = ?/2 < ?.

|f(x) - f(c)| = |(2? + 2)/(? + 2?) - (2? + 2)/?|
= |(2? + 2 - (2? + 2)(? + 2) + (? + 2?))/(? + 2?) - 1|
= |(2? + 2 - 2? - 6? - 4 + ? + 2?)/(? + 2?) - 1|
= |(- ? - 2? - 2)/(? + 2?) - 1|
= |((? + 1) + 1)/(? + 2?) + 1|
> 1.
Hence |f(x)-f(c)| < ? does not hold when ? = 1, contradicting the original supposition.

The question didn't giving the correct definition of continuity.
skipjack is online now  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
1 or x, continuous



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
continuous frankpupu Calculus 4 January 31st, 2012 05:17 AM
Proving that a space is continuous, continuous at 0, and bdd thedoctor818 Real Analysis 17 November 9th, 2010 08:19 AM
continuous map guroten Real Analysis 2 November 5th, 2010 01:46 PM
continuous at 0 summerset353 Real Analysis 2 February 22nd, 2010 03:37 PM
continuous at any point iff continuous at origin babyRudin Real Analysis 6 October 24th, 2008 12:58 AM





Copyright © 2019 My Math Forum. All rights reserved.