I'm asked to prove that, in a version of the Gale-Stewart game where the two players alternately pick zero or one, the pay-off function cannot be continuous. From what i have read, the pay-off function H, typically, assigns the value 1 when the play (sequence of choices) is in the pay-off set A, in which case player 1 wins, and the value 0 when the play is not in A, in which case player 2 wins.
First question: H takes finite sequences, the play/sequence obtained at the point in the game where H is applied, as an argument, right?
Second question/claim: It seems rather obvious that H is not continuous, since its value is either 0 or 1 and it cannot change from one to the other continuously (it reminds me of discrete metric spaces, though they might not be relevant at all). I'm having a bit trouble accepting that it really is that simple, though, so a more formal argument or proof (sketch) would be much appreciated.
(If it is not true that the pay-off function is not continuous, then please say so)