The function equal to $q$ at a rational $p/q$ in lowest terms and to $0$ at every irrational is finite at every point and unbounded on every nondegenerate interval

Let $q(x)$ be the least denominator of a rational $x$ (The Dirichlet function $1_{\mathbb{Q}}$, and Thomae's function $t$ with $t(x) = 1/q$ at a rational $x = p/q$ in lowest terms with $q \ge 1$ and $t(x) = 0$ at every irrational $x$) and define $h : \mathbb{R} \to \mathbb{R}$ by

$h(x) := \iota(q(x)) \ \text{ for } x \in \mathbb{Q}, \qquad h(x) := 0 \ \text{ for } x \notin \mathbb{Q},$

where $\iota(q)$ is the canonical natural (The canonical natural $\iota(n) = n \cdot 1_F$ of a field) and $\mathbb{Q}$ is the canonical copy of the rationals inside $\mathbb{R}$ (The rationals embed densely in the reals). Equivalently $h(x) = 1/t(x)$ at a rational $x$, where $t$ is Thomae's function. Then:

1. $h(x)$ is a real number for every real $x$: $h$ is finite at every point;
2. $h$ is unbounded on every nondegenerate interval (Lower bound, bounded below, bounded set, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length): for all reals $a < b$ and every real $M$ there is $x \in (a,b)$ with $h(x) > M$.

So a function may be finite at every single point and yet fail to be bounded on every interval, however short. In particular $h$ is bounded on no neighbourhood of any point.

• Finiteness at a point says nothing about local boundedness. The two notions are often conflated; this example separates them as sharply as possible, since $h$ is unbounded on every nondegenerate interval and yet takes only real values.

• $h$ is nowhere continuous, and the oscillation is infinite everywhere. Continuity at $c$ would give a neighbourhood on which $|h - h(c)| < 1$, hence a neighbourhood on which $h$ is bounded, which claim 2 forbids; equivalently $\omega_{h}(c) = +\infty$ at every real $c$ (The oscillation $\omega_f(S) = \sup\{\,|f(x) - f(y)| : x, y \in S\,\}$ of $f$ on a set and the oscillation $\omega_f(c) = \inf_{\delta > 0} \omega_f(A \cap N_\delta(c))$ at a point, both taken in the extended reals, Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point). This is a different pathology from Thomae's function, whose values are bounded by $1$ and which is continuous at every irrational (The Dirichlet function is continuous at no point of $\mathbb{R}$, and Thomae's function is continuous at every irrational and at no rational, so its set of continuity points is exactly the set of irrationals and its oscillation at $c$ equals $t(c)$).

• The separation estimate is the only arithmetic used. Step 1.2 is the standard fact that two distinct rationals with denominators $q_{1}$ and $q_{2}$ are at least $1/(q_{1}q_{2})$ apart, and it is proved from nothing more than "a nonzero integer has absolute value at least $1$".