> > I think you need a powerset axiom to formally construct the set of functions mapping a given set to another -- e.g. the set of continuous functions on the reals. Isn't that important to be able to do?
Some might believe that physicists (for example) don't need continuous functions on the reals. To which some might reply that it depends what you mean by 'need'. Perhaps there are no physical phenomena that are continuous functions on the reals*, but there are things that (to date, at least) are best treated as if they were continuous functions on the reals.
* I have no opinion on that myself.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting