Limits of a monotone function

Question

Let f: $\Bbb R \mapsto \Bbb R $ be a continuous function, we know that f is strictly decreasing. Can we directly say that $\lim_{x\to\infty} f = -\infty$ ?

Answer 1

If $f(\mathbb R)=\mathbb R$, then yes. Otherwise, now, there are many functions that are strictly decreasing and have a lower bound. Simple examples would be $f(x)=e^{-x}$, or $f(x)=-\arctan(x)$