Thomae function

Thomae's function 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

Theorem: The Thomae function is continuous at all irrational numbers and discontinuous at all rational numbers.

Proof: █

Theorem: The Thomae function has a (strict) local maximum at each rational number.

Proof: █

Theorem: The Thomae function $f(x)$ is not Riemann integrable but it is Lebesgue integrable and $$\displaystyle\int_0^1 f(x) dx = 0.$$

Proof: █