If you pick any topology on Y, then there is a smallest topology on X such that the gi are all continuous (the one generated by all preimages of open sets in Y under the gi). As Cameron points out, there may then be extra continuous functions between X and Y. What isn't immediately clear to me (but I'd be interested to know) is what conditions you need on the set {gi} such that this doesn't happen; this could easily depend on the choice of topology for Y. Alternatively, are there conditions on the gi giving a unique topology on X and Y so they're continuous.