= Solution
Use the <reduced-denominator function unbounded on every interval>:
$$
\boxed{f_4(x)=\begin{cases}q,&x=p/q\text{ in lowest terms},\ q>0,\\0,&x\notin\mathbb Q.\end{cases}}
$$
Every value is a finite <real number>. In any nondegenerate interval $[a,b]\subseteq[0,1]$, its interior contains infinitely many <rational numbers>, whereas there are only finitely many reduced fractions in $[0,1]$ with denominator at most any fixed $M$. Some <rational number> in $(a,b)$ therefore has denominator exceeding $M$. Hence the <function> is unbounded on every such interval. It reverses the positive values of the <Thomae function> rather than requiring an infinite value at any point.
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