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

to be

at reduced rationals

and

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

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

on the rationals and

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