Exercise 7.3.2

Question

Recall that Thomae's function
$$
t(x)= \begin{cases}1 & 	ext { if } x=0 \ 1 / n & 	ext { if } x=m / n \in \mathbf{Q} \backslash\{0\} 	ext { is in lowest terms with } n>0 \ 0 & 	ext { if } x 
otin \mathbf{Q}\end{cases}
$$
has a countable set of discontinuities occurring at precisely every rational number. Follow these steps to prove $t(x)$ is integrable on $[0,1]$ with $\int_{0}^{1} t=0$.
\begin{enumerate}[label=(\alph*)] 
\item First argue that $L(t, P)=0$ for any partition $P$ of $[0,1]$.
\item Let $\epsilon>0$, and consider the set of points $D_{\epsilon / 2}=\{x \in[0,1]: t(x) \geq \epsilon / 2\}$. How big is $D_{\epsilon / 2}$ ?
\item To complete the argument, explain how to construct a partition $P_{\epsilon}$ of $[0,1]$ so that $U\left(t, P_{\epsilon}ight)<\epsilon$.
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item Because the irrationals are dense, for any interval there must be some point in that interval where $t(x) = 0$, which is also the minimum possible value of $t$; hence $L(t,P) = 0$.
\item This is the size of the set of rational numbers which, when expressed in lowest terms as $m/n$, have $n \leq 2/\epsilon$. This must be finite; let the size of $D_{\epsilon/2}$ be denoted as $|D_{\epsilon/2}|$.
\item Informally, we'll set up the partition to ``isolate'' each point in $D_{\epsilon/2}$. Define the partition as
$$P_\epsilon = \{d + k/2 : d \in D_{\epsilon/2}\} \cup \{d - k/2 : d \in D_{\epsilon/2}\} \cup \{0,1\}$$
where $k = \epsilon / (2 |D_{\epsilon/2}|)$. (If some of these isolating intervals overlap, then decrease $k$ until they don't; this can be formalized by requring $k$ be the minimum of what it is currently and the smallest distance between two points in $D_{\epsilon/2}$.) Then
$$U(t,P_\epsilon) < \epsilon/2 + k |D_{\epsilon/2}| = \epsilon$$
where the first term comes from all partition intervals that aren't isolating an element in $D_{\epsilon/2}$, and the second term comes from the elements in $D_{\epsilon/2}$.
 \end{enumerate}

Exercise 7.3.2

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Exercise 7.3.2

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