So you are asking: which topological spaces, besides the discrete ones, are such that for every strictly positive real function $g$ there is a strictly positive continuous real function $f$ with $0<f<g$? Only some hints.
Others have already noted that there cannot be nontrivial convergent sequences.
You can note that if there are no nontrivial convergent sequences, then replace $g$ with the largest $h<g$ which takes only values of the form $1/n$, and then note that this $h$, even if not continuous, at least does not give the above noted problem (forcing a possible continuous $f$ to have value $0$). Taking the closures of the sets where $h>1/n$ one can then use Urhyson's lemma / Tietze extension theorem (when the space is normal) and so obtain a continuous $f$ with $0<f<h$.
Is there a non-discrete normal space with no nontrivial converging sequence? You can easily find the answer in any standard book on general topology which treats Stone compactification.
