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Checking for integrability on a half-open interval

Problem Statement
For a bounded, continuous and monotonous function on a half-open interval ##(a,b]##, how does one check if the function is integrable? (specifically Darboux integrable)
Relevant Equations
My definition of Darboux integrable: ##U(f,P)-L(f,P)<\epsilon## for all ##\epsilon>0##
For a closed interval ##[a,b]## I have learned that ##U(f,P)-L(f,P)=\frac{(f(b)-f(a))\cdot(b-a)}{N}## where ##N## is the number of subintervals of ##[a,b]## (if ##f## is monotonically decreasing, change the numerator of the fraction to ##f(a)-f(b)##). However, if the interval is half-open, then ##f(a)## is no longer defined. How does one go about this issue?
 

PAllen

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I assume you mean monotonic, not monotonous, though a dubious case can be made for synonimity.
 

PAllen

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More seriously, I assume you have been presented with a definition of the integral for cases other than monotonic. Try working from that definition.
 
Last edited:
Yes, I mean monotone of course. The only definition of Darboux integrability I’ve been given is the epsilon one above, although this can be written somewhat differently. And when I check the integrability for a certain bounded, continues and monotone function on a closed interval I make use of the fact that the difference between the upper and lower Riemann sums create a column of height ##f(b)-f(a)## and width ##(b-a)/N##. Then finding an ##N## for all ##\epsilon>0## is quite easy.
 

PAllen

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Just think about how to extend the argument slightly. You have monotonicity and continuity given.
 
Could one write ##\lim\limits_{x \to a^+}f(x)## instead of ##f(a)##? Is this limit evaluable?
 

pasmith

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Problem Statement: For a bounded, continuous and monotonous function on a half-open interval ##(a,b]##, how does one check if the function is integrable? (specifically Darboux integrable)
Relevant Equations: My definition of Darboux integrable: ##U(f,P)-L(f,P)<\epsilon## for all ##\epsilon>0##

For a closed interval ##[a,b]## I have learned that ##U(f,P)-L(f,P)=\frac{(f(b)-f(a))\cdot(b-a)}{N}## where ##N## is the number of subintervals of ##[a,b]## (if ##f## is monotonically decreasing, change the numerator of the fraction to ##f(a)-f(b)##). However, if the interval is half-open, then ##f(a)## is no longer defined. How does one go about this issue?
Don't use that formula: it assumes a closed interval and is therefore not appropriate to a half-open interval.

In fact, [itex]f[/itex] will be Darboux integrable on [itex](a, b][/itex] because it is bounded and continuous; it is not necessary that it be monotonic (for example, [itex]\sin (x^{-1})[/itex] is integrable on [itex](0,1][/itex]).

The starting point is the definition of Darboux integrability:

A function [itex]f : (a,b] \to \mathbb{R}[/itex] is integrable if and only if for every [itex]\epsilon > 0[/itex] there exists a partition [itex]P[/itex] such that [itex]U(f,P) - L(f,P) < \epsilon[/itex].

The key observation is that as [itex]f[/itex] is bounded the contribution to [itex]U(f,P) - L(f,P)[/itex] from the first subinterval is bounded above by [itex]K\delta[/itex] where [itex]\delta[/itex] is the width of the subinterval and [itex]K \geq 0[/itex] does not depend on the choice of partition. This can be made arbitrarily small by suitable choice of [itex]\delta[/itex]. As [itex]f[/itex] is continuous and thus integrable on the closed bounded interval [itex][a + \delta, b][/itex], the contribution to [itex]U(f,P) - L(f,P)[/itex] from [itex][a + \delta, b][/itex] can be made arbitrarily small by suitable choice of subintervals.
 
The key observation is that as [itex]f[/itex] is bounded the contribution to [itex]U(f,P) - L(f,P)[/itex] from the first subinterval is bounded above by [itex]K\delta[/itex] where [itex]\delta[/itex] is the width of the subinterval and [itex]K \geq 0[/itex] does not depend on the choice of partition. This can be made arbitrarily small by suitable choice of [itex]\delta[/itex]. As [itex]f[/itex] is continuous and thus integrable on the closed bounded interval [itex][a + \delta, b][/itex], the contribution to [itex]U(f,P) - L(f,P)[/itex] from [itex][a + \delta, b][/itex] can be made arbitrarily small by suitable choice of subintervals.
What is ##K##? If one were to prove that a piece-wise continuous function is integrable, say ##f(x)=\tan{(x)}## on ##[0,\pi/3)## and ##f(x)=\sin{(x)}## on ##(\pi/3,\pi]##, does one simply show that we can treat both intervals as closed intervals?
 

PAllen

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What does bounded mean? That should give you a clue to what K might be.
 

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