This question asks us to show that $\Bbb R$ with the following metric is not complete:
Fix a strictly positive function $f \in L^1(\Bbb R)$, and let $d(x,y)=\left|\int_x^y f(t)dt\right|$.
It's easy to see (in line with the answers to the linked question) that any unbounded increasing sequence of real numbers is Cauchy without limit in this metric. What is the completion of this metric space? Is it homeomorphic to any well-known space? If not, what are some interesting properties it has? If the problem is intractable for arbitrary $f$, feel free to choose a suitably interesting $f$ or subclass of such $f$. (In particular, continuous $f$ or smooth $f$.)