陶哲轩实分析（上）11.5及习题-Analysis I 11.5

Exercise 11.5.1

Since $f$ is piecewise continuous, we can find a parititon $\mathbf P$ such that $f|_J$ is continuous on $J$ for $∀J∈\mathbf P$. By Proposition 11.5.3, we have $f|_J$ is Riemann integrable on $J$.
We define
$F_J (x)=\begin{cases}f|_J (x),&x∈J\\0,&x∈I\backslash J\end{cases}$
By Theorem 11.4.1(g), $F_J$ is Riemann integrable on $I$, and we further have
$f(x)=\sum_{J∈P}F_J (x)$
So by Theorem 11.4.1(a), $f$ is Riemann integrable on $I$.