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.