We prove the existence of a subgame-perfect e-equilibrium, for every e > 0, in a class of multi-player games with perfect information, which we call free transition games. The novelty is that a non-trivial class of perfect information games is solved for subgame-perfection, with multiple non-terminating actions, in which the payoff structure is generally not (upper or lower) semi-continuous. Due to the lack of semi-continuity, there is no general rule of comparison between the payoffs that a player can obtain by deviating a large but finite number of times or, respectively, infinitely many times. We introduce new techniques to overcome this difficulty.our construction relies on an iterative scheme which is independent of e and terminates in polynomial time with the following output: for all possible histories h, a pure action ah1 or in some cases two pure actions ah2 and bh2 for the active player at h. The subgame-perfect e-equilibrium then prescribes for every history h that the active player plays ah1 with probability 1 or respectively plays ah2 with probability 1 - d(e) and bh2 with probability d(e). Here, d(e) is arbitrary as long as it is positive and small compared to e, so the strategies can be made “almost” pure.