In existing game models, total functionals have no simple characterization neither in term of game strategies, nor in term of the total set-theoretical functionals they define. We show that the situation changes if we extend the usual notion of game by allowing infinite plays. Total functionals are, now, exactly those having a tree-strategy in which all branches end in a last move, winning for the strategy. Total functionals now define (via an extensional collapse) all set-theoretical functionals. Our model is concrete: we used infinite computations only to have a nice characterization of totality. A computation may be infinite only when the input is a discontinous functional; in practice, never.