(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let f(x)= { 1 if x=[tex]\frac{1}{n}[/tex] for some n[tex]\in[/tex] the natural numbers,

or 0 otherwise}

Prove f is integrable on [0,1], and evaluate the integral.

2. Relevant equations

This is using Riemann Integrability. I know that the method of providing the solution is supposed to be by application of the following theorem:

"A bounded function f is integrable on [a,b] if and only if [tex]\forall \epsilon > 0[/tex], there is a partition [tex]P_{\epsilon}[/tex] of [a,b] where U(f, [tex]P_{\epsilon}[/tex]) - L(f,[tex]P_{\epsilon} < \epsilon[/tex]"

U([tex]P_{\epsilon}[/tex]) and L([tex]P_{\epsilon}[/tex]) are of course the upper and lower sums.

3. The attempt at a solution

I know that L([tex]P_{\epsilon}[/tex]) is 0 everywhere, because for any sub-interval no matter how small there is some value of x where x /= [tex]\frac{1}{n}[/tex]. I also know that the integral must equal 0, as the lower and upper sums must be equal in an integrable function.

What I am supposed to do is to find some partition of [a,b] where the upper sum is less than epsilon, but I can't figure out how to do that.

Given an epsilon, if I select some x_{0}where x_{0}=[tex]\frac{1}{m}[/tex] for some m and x_{0}< [tex]\epsilon[/tex], then it follows that the upper sum over [0,x_{0}] < [tex]\epsilon[/tex]. The issue is then the upper sum on [x_{0}, 1].

I have also noticed this fact: Suppose my x_{0}=[tex]\frac{1}{500}[/tex]. Then if I move up to [tex]\frac{2}{500}[/tex], that's actually [tex]\frac{1}{250}[/tex]. Similarly, [tex]\frac{4}{500}=\frac{1}{125}[/tex], [tex]\frac{5}{500}=\frac{1}{100}[/tex], and [tex]\frac{10}{500}={\frac{1}{50}[/tex]. That means that between [tex]\frac{2}{500}[/tex] and [tex]\frac{1}{50}[/tex], there are only 4 values equal to some [tex]\frac{1}{n}[/tex]. The upper sum over that would then be simply 4 * [tex]\frac{1}{500}[/tex]. Somewhere in all of that, I feel like there is a way to partition [x_{0}, 1] so that it's less than a given epsilon, but I can't figure it out.

Thanks in advance.

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# Homework Help: Prove Integrability of a Discontinuous Function

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