Exercise 7.6.1

Recall that Thomae’s function

has a countable set of discontinuities occurring at precisely every rational number. Let’s prove that Thomae’s function is integrable on $[0,1]$ with ${\int }_0^{1}t=0$.

(a)

First argue that $L(t,P)=0$ for any partition $P$ of $[0,1]$.

(b)

Let $𝜖>0$, and consider the set of points $D_{𝜖∕2}=\{x\in [0,1]:t(x)\ge 𝜖∕2\}$. How big is $D_{𝜖∕2}$ ?

(c)

To complete the argument, explain how to construct a partition $P_{𝜖}$ of $[0,1]$ so that $U\left(t,P_{𝜖}\right)<𝜖$.