Hi,

I don't think you've stated your question correctly. Given what you said:

Let f be a continuous function with domain S (subset of R) and image S' = f(S). If S' is an interval, then S is an interval.

This is clearly false:

Let and f(x)=x^{2}. Then f(S)=[1,4], but S is not an interval.

Did you omit the hypothesis that f is one to one? But if so, the problem is trivial as you point out.