Example: Continuous function having a limit along every arithmetic sequence but having no limit over the reals The following question seems to be very elementary and must be a folklore, but we are not able to find an answer.

Let $f: [0,\infty)\to \mathbb R$ be a continuous function such that for every $a&gt;0$,
$$
\lim_{n\to+\infty} f(na) = 0,
$$
where $n$ is restricted to be integer. 

Does this imply that 
$$
\lim_{t\to+\infty} f(t) = 0,
$$
where $t$ is allowed to be real?
Yes. The important thing here is that $f$ is continuous so if limit on discrete variable is zero, limit on continuous variable is also zero.
Yes. The important thing here is that $f$ is continuous so if limit on discrete variable is zero, limit on continuous variable is also zero.