ruler function

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

f(x)=\left\{\begin{array}[]{ll}0,&\hbox{$x$ is irrational;}\\ 1/n,&\hbox{$x=m/n$, $m$ and $n$ are relatively primes}.\end{array}\right.

(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.

References

1 Dunham, W., Nondifferentiability of the Ruler Function, Mathematics Magazine, Mathematical Association of America, 2003.
2 Heuer, G.A., Functions Continuous at the Irrationals and Discontinuous at the Rationals, The American Mathematical Monthly, Mathematical Association of America, 1965.

Title	ruler function
Canonical name	RulerFunction
Date of creation	2013-03-22 18:23:55
Last modified on	2013-03-22 18:23:55
Owner	yesitis (13730)
Last modified by	yesitis (13730)
Numerical id	5
Author	yesitis (13730)
Entry type	Definition
Classification	msc 26A99