Thomae_function

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Definition

The Thomae function (or modified Dirichlet function) $t: \mathbb{R} \to \mathbb{R}$ is defined by

$$ t(x) = \begin{cases} 1 & \text{if } x = 0 \\ 1/n & \text{if } x = m/n \in \mathbb{Q} \text{ in lowest terms with } n > 0 \\ 0 & \text{if } x \notin \mathbb{Q} \end{cases} $$

Proof of Continuity Properties

Claim: $t$ is continuous at every irrational point and discontinuous at every rational point.

Continuous at irrationals: Let $c \notin \mathbb{Q}$ and $\epsilon > 0$. Choose $N$ with $1/N < \epsilon$. There are only finitely many rationals $m/n$ in lowest terms with $n \leq N$ near $c$. Choose $\delta$ to avoid these.

For $|x - c| < \delta$: if $x \notin \mathbb{Q}$, then $|t(x) - t(c)| = 0$. If $x = m/n$ with $n > N$, then $|t(x) - t(c)| = 1/n < \epsilon$.

Discontinuous at rationals: Every neighborhood of a rational contains irrationals where $t = 0$, but $t(c) > 0$ for $c \in \mathbb{Q}$.