$\lim_n{f(\frac{x}{n})}=0$, for all $x\in (0,1)$ implies $lim_{x\to0}f(x)=0$ (edit, $f$ is not continuous)
If $f(x/n)\to0$ when $n\to\infty$, for every $x$ and $f$ is continuous, then $f(x)\to0$ when $x\to0$, or not?