Variations and extensions of Croft's problem

In this work we study the following classical still challenging Calculus problem: {\it If $f:(0,\mathrm{\infty })\to \mathbb{R}$ is a continuous function, for which the sequence $\{f(nx)\}$ tends to zero, for every positive $x$, as $n$ tends to infinity, then $f(x)$ also tends to zero, as $x$ tends to infinity.}