Well, is the set of allboundedfunctions continuiously. In your case you just lucked out that continuity implies boundedness.

I'm not sure what you mean here? You can turn into a ring?-The group of units is all continuous functions that are invertible on [0,1].

By my understanding: those that have no asymptotes (continuity), and no roots (invertibility) on [0,1].

Is there any way to classify these functions more rigourously?

Thanks