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.
Added: please replace "no nontrivial convergent sequences" with the stronger "every countable subset is closed". And examples are even more exotic, but Gillman and Jerison, rings of continuous functions, should have them (and perhaps also Engelking).
Concerning TeX commands: I do not use them on purpose, but if it this the rule to always use them here, then I will in future only give answers that do not require mathematical notation, so that we all will be able to read in the way we like.
