Thomae function

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Thomae's function (sometimes called the popcorn function, raindrop function, Stars over Babylon) is given by the formula $$f(x) =\begin{cases} 1 & \text{if } x= 0 \\ \tfrac1{q} & \text{if } x = \tfrac{p}{q}\\ 0 & \text{if } x \in \mathbb{R}-\mathbb{Q}. \end{cases}$$


Thomae function is continuous at irrationals
Thomae function is discontinuous at rationals


Thomae Function by Bret Benesh (11 January 2012)
Thomae Function by Douglas Harder (19 April 2012)

See also[edit]

Modifications of Thomae's Function and Differentiability