Exercise 7.10

The discontinuities of the $n^{\mathit{th}}$ term, $f_n(x)=(\mathit{nx})∕n^2$, occur at the rational numbers $m∕n$ for any integer $m$. Let $y=p∕q$ be a rational number where $p$ and $q$ have no common divisors. Then $f_n$ is discontinuous at $y$ if and only if $n$ is a multiple of $q$. Let $g_q={\sum <!--\; FUNCTION\; APPLICATION\; -->}_m(\mathit{mqx})∕{(\mathit{mq})}^2$, which is discontinuous at $y$, and let $h_q=f-g_q$. Then the partial sums of $h_q$ are all continuous at $y$, hence by Theorem 7.11, $h_q$ is also continuous at $y$, and so $f=g_q+h_q$ is discontinuous at $y$. If the real number $z$ is irrational, then the partial sums of $f$ are all continuous at $z$, hence by again applying Theorem 7.11 we have $f$ is continuous at $z$. Therefore the discontinuities of $f$ are the rational numbers, a countable dense subset of the real numbers.

The partial sums of $f$ converge uniformly to $f$ and are all Riemann-integrable on bounded intervals. Hence by Theorem 7.16, $f$ is also Riemann-integrable on bounded intervals.