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 / Tieze extension theorem (when the space is normal) and so obtain a continuos 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.