Prove that the inverse image of an analytic pointset by a continuous function is analytic.

Hint. Aim for an equivalence of the form where is continuous and are defined by , and then use the result that says:

If are continuous functions, then the set of points on which they agree is closed.

I am not sure how to prove this. I would appreciate a few hints or suggestions. In our book, denotes the Baire space. Also, subsets of the Baire space are called pointsets. Thanks in advance.