Chapter 2, Problem 18AAE

The Dirichlet ruler function If $x$ is a rational number, then $x$ can be written in a unique way as a quotient of integers $m/n$, where $n>0$ and $m$ and $n$ have no common factors greater than 1. (We say that such a fraction is in lowest terms. For example, $6/4$ written in lowest terms is $3/2$.) Let $f\left(x\right)$ be defined for all $x$ in the interval $\left[0,1\right]$ by

$f\left(x\right)=\{\begin{array}{l}1/n\text{ if }x=m/n\text{ is a rational number in lowest terms}\\ \text{0 if }x\text{ is irrational}\text{. }\end{array}$

For instance, $f\left(0\right)=f\left(1\right)=1,f\left(1/2\right)=1/2,f\left(1/3\right)=f\left(2/3\right)=1/3,$ $f\left(1/4\right)=f\left(3/4\right)=1/4$, and so on.

1. Show that $f$ is discontinuous at every rational number in $\left[0,1\right]$.

2. Show that $f$ is continuous at every irrational number in $\left[0,1\right]$. (Hint: If $\epsilon$ is a given positive number, show that there are only finitely many rational numbers $r$ in $\left[0,1\right]$ such that $f\left(r\right)\ge \epsilon$.)

3. Sketch the graph of $f$. Why do you think $f$ is called the “ruler function”?