Definition:Thomae Function

Definition

Thomae Function on $\openint 0 1$

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 Johann Peter Gustav Lejeune Dirichlet
• the (small) Riemann function after Georg Friedrich Bernhard Riemann
• 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 John Horton Conway)

Also see

• Definition:Dirichlet Function
• Definition:Ruler Function

• Results about the Thomae function can be found here.

Source of Name

This entry was named for Carl Johannes Thomae.

Sources

• Weisstein, Eric W. "Dirichlet Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletFunction.html