June 16th, 2019, 09:22 PM  #1 
Member Joined: Apr 2017 From: India Posts: 73 Thanks: 0  Integrability
Is an increasing/decreasing but noncontinuous 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. 
June 17th, 2019, 12:28 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,823 Thanks: 723 
In general yes, but be careful of infinities, either in the function or the domain of integration.

June 17th, 2019, 12:52 PM  #3 
Senior Member Joined: Aug 2012 Posts: 2,395 Thanks: 750  
June 17th, 2019, 04:48 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 648 Thanks: 412 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. 
June 17th, 2019, 11:18 PM  #5 
Senior Member Joined: Oct 2009 Posts: 867 Thanks: 330 
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. 
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 noncontinuous, 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. 

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