Reduced-denominator function unbounded on every interval ($f(p/q)=q,f(x)=0(x\notin \mathbb{Q})$)

On $[0,1]$, use reduced representations of rational numbers with positive denominator. This reverses the nonzero values of the Thomae function, while leaving its values at irrational numbers zero. Every interval of positive length contains rational numbers of arbitrarily large reduced denominator: there are only finitely many fractions in $[0,1]$ with denominator at most a fixed $M$, whereas any open interval contains infinitely many rational numbers. Thus this function taking finite real number values is unbounded on every nondegenerate subinterval.