Thomae function

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

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  Plot of the Thomae function.

Properties

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

Videos

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

See also

Modifications of Thomae's Function and Differentiability

References

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