The Popcorn Function

Problem 1. Prove that the popcorn function by $f : [0, 1] \to \mathbb R$ by

$\displaystyle f(x) = \begin{cases} 1/q, & x=p/q,\ p,q\ \text{coprime,} \\ 0, & \text{otherwise}. \end{cases}$

is Riemann-integrable.

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Solution. Fix $\epsilon > 0$ . The key insight is to observe that for any $k > 0$ , there are only finitely many rationals $x$ such that $1/f(x) < 1/(k \cdot \epsilon) \iff f(x) > k \cdot \epsilon$ . Denote these rationals in ascending order by $\{r_1,\dots,r_M\}$ . To “separate” these rationals, define

$\displaystyle \delta := \frac 13 \cdot \min_{2 \leq i \leq M} (r_i - r_{i-1})$

Define the partition

$\displaystyle P := \{0,1\} \cup \bigcup_{i=1} \{r_i - \delta, r_i + \delta\} \subseteq [0, 1].$

Then

$\begin{aligned} U(f, P) - L(f, P) &= \sum_{i = 1}^{|P|} (M_i(f, P) - m_i(f, P)) \Delta x_i \\ &\leq (1 - 0) \cdot (k \cdot \epsilon) + M \cdot 1 \cdot 2\delta\\ &= k \cdot \epsilon + 2M \cdot \delta. \end{aligned}$