ruler function

The ruler function $f$ on the real line is defined as follows:

(1)

Given a rational number $\frac{m}{n}$ in lowest terms, $n$ positive, the ruler function outputs the size (length) of a piece resulting from equally subdividing the unit interval into $n$, the number in the denominator, parts. It “ignores” inputs of irrational functions, sending them to 0.

The ruler function is so termed because it resembles a ruler. The following picture might be helpful: if $\frac{m}{n}$ in lowest terms is a reasonably small rational number (which we assume positive). Then it can be “read off” on a ruler whose intervals of one unit size are each equally subdivided into $n$ parts measuring $\frac{1}{n}$ units each by

1. running one’s finger through until the integer preceding it and then

2. running through to the subsequent $r$th subunit, “left-over” from the division of $m$ by $n$.

On the other hand, an irrational number can not be read off from any ruler no matter how fine we subdivide a unit interval in any ruler.