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Let $h(x)= 0$ if $x$ is irrational and $1/q$ if $x=p/q$ where $p,q$ are natural numbers such that $gcd(p,q)=1$.

Show that the function is Riemann integrable.

The first part of the proof says: Let $\epsilon>0$ let $N\in \mathbb{N}$ such that $\frac{1}{N}<\frac{\epsilon}{2}$, then the number of $x's$ such that $h(x)>\frac{1}{N}$ are at most $N^2$.

Then for any partition $P$ of $[0,1]$ there are at most $2N$ intervals containing some $x$ such that $h(x)>\frac{1}{N}$.

My question is, how do you prove the statement in bold? and what's the intuition for it? Lastly, when would I use such a method to show a function is integrable?

Semiclassical
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  • The worst case, when it's exactly $2N$, is when the partitions go from $0$ to $1/(2N)$, from $1/(2N)$ to $1/N$, from $1/N$ to $3/(2N)$, etc. – wlad Dec 07 '19 at 21:56
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    This has been addressed many times on this site. See here for an answer to your specific question and the follow-up with slight notational differences. – RRL Dec 07 '19 at 21:57

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