When can a function be said to be bounded

Question

If $f$ is definied on $(0, ∞)$ -> $(0, ∞)$ and is continuous and decreasing. Is it possible to conclude from this information that $f$ is bounded?

My intuition tells me that $f$ should be bounded since it is continuous and decreasing, but since it is defined on an open interval it is giving me some concern since the boundedness theorem (http://mathonline.wikidot.com/boundedness-theorem) says that the function needs to be defined on a closed interval for $f$ to bounded.

Any help would be greatly appreciated.

Thanks.

Answer 1

What about $f(x) = \frac{1}{x}$?