Let $X:=\{f: \mathbb{C}\to \mathbb{C}\}$ be a class of *total* functions on $\mathbb{C}$ closed under composition, addition, multiplication, and scalar multiplication. Does there exist a topology on $\mathbb{C}$ making these functions and only these functions continuous?

If it's not true in general (it probably isn't), are there any interesting known cases where it is true?

Note: I emphasize *total* functions because we want them to be everywhere defined. This avoids functions with bad singularities.

Edit: Obviously, continuous functions in the standard topology fit this bill, but this is tautological and not in the spirit of the problem.

Edit 2: Apparently the way I asked this question made it seem like I was looking for an answer to the "general case" which seems pretty untrue although I haven't actually worked it out. Rather, the real question was interesting cases where it is true.