Are there a sequence $\{a_n\}$ and a continuous function $f$ such that $\lim_{n\to\infty} f(a_n) \in \mathbb{R}$ but $\{a_n\}$ diverges.

Question

If $f$ is continuous at $\alpha$ and $\lim_{n\to\infty} a_n = \alpha$, then $\lim_{n\to\infty} f(a_n) = f(\alpha)$.

Are there a sequence $\{a_n\}$ and a continuous function $f$ such that $\lim_{n\to\infty} f(a_n) \in \mathbb{R}$ but $\{a_n\}$ diverges.

Answer 1

I found an example now. Sorry.
$f(x) = \sin(x)$ and $a_n = \pi n$.