**Definition**: Thomae's function is defined $f: (0,1) \to \mathbb{R}$ such that $f(x) = \begin{cases} \frac{1}{q} & x=\frac{p}{q} \text{ in lowest form, } p\geq 0 \\ 0 & x \not\in \mathbb{Q}. \end{cases}$ **Graph**![[Graph of Thomae's function.png]] **Remarks**: - $f$ is discontinuous at all rational numbers, - $f$ is [[Continuous Function (Epsilon-Delta Definition)|continuous]] at all irrational numbers. ### Theorem - [[Thomae's function is discontinuous at rational points and continuous at irrational points]] -