The Modified Dirichlet Function

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.


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