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 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.
 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.
June 17th, 2019, 12:52 PM   #3
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 Originally Posted by mathman 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.

 June 17th, 2019, 04:48 PM #4 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
 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.
 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.

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