My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Thanks Tree1Thanks
  • 1 Post By SDK
Reply
 
LinkBack Thread Tools Display Modes
June 16th, 2019, 09:22 PM   #1
Member
 
Joined: Apr 2017
From: India

Posts: 73
Thanks: 0

Integrability

Is an increasing/decreasing but non-continuous function always integrable?

This question requires some sort of counter example, if it is incorrect. I am unable to get the intuitive idea so as how to answer this question. Please help.
shashank dwivedi is offline  
 
June 17th, 2019, 12:28 PM   #2
Global Moderator
 
Joined: May 2007

Posts: 6,786
Thanks: 708

In general yes, but be careful of infinities, either in the function or the domain of integration.
mathman is offline  
June 17th, 2019, 12:52 PM   #3
Senior Member
 
Joined: Aug 2012

Posts: 2,342
Thanks: 731

Quote:
Originally Posted by mathman View Post
In general yes, but be careful of infinities, either in the function or the domain of integration.
Proof or counterexample please. Is the Riemann hypothesis true? "In general yes, but be careful of infinities." Give that man his million dollars.
Maschke is online now  
June 17th, 2019, 04:48 PM   #4
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 635
Thanks: 401

Math Focus: Dynamical systems, analytic function theory, numerics
It isn't even true for continuous functions so I'm not sure why you think it should be true if you allow discontinuous functions.

For example, $f(x) = x$ defined on $\mathbb{R}$ is not integrable. Now redefine its value at a point so that it isn't continuous and it is still not integrable.
Thanks from topsquark
SDK is offline  
June 17th, 2019, 11:18 PM   #5
Senior Member
 
Joined: Oct 2009

Posts: 841
Thanks: 323

It depends on your domain of definition really.
If you work on a closed interval and work with Riemann integration, then yes, every monotone function is integrable.
This follows either directly through some manipulations of the Riemann sums. Or you can show a monotone function has countably many discontinuities, this implies Riemann integrability through the (difficult) Lebesgue theorem.
Micrm@ss is offline  
June 18th, 2019, 01:39 AM   #6
Member
 
Joined: Apr 2017
From: India

Posts: 73
Thanks: 0

If a monotone function is bounded, then it will be Riemann integrable. It doesn't matter whether it is continuous or not.

I meant if the function is continuous throughout the domain, then also it will be Riemann integrable and and even if it is discontinuous at countable number of points, then also it will be integrable.

Did I get that right?

Also, suppose there is a function(we don't know whether it is monotonic or not, let's say, in general) that is bounded but it is non-continuous, then Can I say that, we can't comment on the Riemann integrability of that function in general and therefore it will vary according to whether what is the definition of the domain.

Last edited by shashank dwivedi; June 18th, 2019 at 01:45 AM.
shashank dwivedi is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
integrability



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Integrability problem Azzajazz Calculus 3 May 19th, 2015 03:44 AM
integrability Sambit Calculus 5 January 22nd, 2011 03:09 PM
help prove integrability kelvinng Real Analysis 1 January 31st, 2009 07:21 PM
help on proving integrability kelvinng Calculus 2 January 28th, 2009 01:30 PM
help..prove integrability kelvinng Real Analysis 1 January 20th, 2009 06:37 PM





Copyright © 2019 My Math Forum. All rights reserved.