Example: Continuous function having a limit along every arithmetic sequence but having no limit over the reals

Question

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 $a>0$, $$ \lim_{n\to+\infty} f(na) = 0, $$ 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?

Answer 1

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.