I vaguely remember studying these as well as one that was something like: h(a/b) = b for rationals a/b (in smallest terms), 0 otherwise This one has the prope

I vaguely remember studying these as well as one that was something like: h(a/b) = b for rationals a/b (in smallest terms), 0 otherwise This one has the properties you mention of the Dirichlet function whilst also being unbounded in every non-empty interval of the reals.