Definition:Thomae Function

Definition

 * Thomae-function.png

The Thomae function $D_M: \R \to \R$ is the real function defined as:
 * $\forall x \in \R: \map {D_M} x = \begin {cases} 0 & : x = 0 \text { or } x \notin \Q \\ \dfrac 1 q & : x = \dfrac p q : p \perp q, q > 0 \end {cases}$

where:
 * $\Q$ denotes the set of rational numbers
 * $p \perp q$ denotes that $p$ and $q$ are coprime (that is, $x$ is a rational number expressed in canonical form)

Also known as
The Thomae function is also seen styled as Thomae's Function.

It has several names in the literature:


 * the modified Dirichlet function after
 * the (small) Riemann function after
 * the popcorn function
 * the raindrop function
 * the countable cloud function
 * the ruler function (although strictly speaking the ruler function is a restriction of this to the dyadic rationals)
 * Stars over Babylon (coined by )

Also see

 * Definition:Dirichlet Function
 * Definition:Ruler Function