The Modified Dirichlet Function

maximum denominator

20

choose

graph

spikes

triangles

This modified Dirichlet function has many names: Thomae, Riemann, popcorn, raindrop, ruler. It is defined on the closed interval

[0,1]

to be

1/q

at reduced rationals

p/q

and

0

elsewhere. It has the curious property that it is continuous on the irrationals but discontinuous at every rational in

(0,1)

.

In contrast, the Dirichlet function (not shown here) is defined to be

1

on the rationals and

0

on the irrationals. It is discontinuous everywhere and its dull graph consists of two blurry lines.