In two-player zero-sum games, for every ε > 0, ε-optimal strategies can be selected Lipschitz continuously as the payoffs vary. For general two-player games this fails as strongly as possible: for any ε > 0, there is no continuous map assigning to each game a Nash ε-equilibrium. We quantify this obstruction by the cost of continuity of a family of games, defined as the threshold above which the family admits a continuous selection of Nash ε-equilibria. Already for symmetric 2×2 games with payoffs in [0,1], the maximal cost is 1/17 over line segments and 1/8 over paths, and the obstruction persists for common-interest games and grows with more players. A parallel contrast holds for discounted stochastic games. In zero-sum games, ε-optimal strategies can be selected Lipschitz continuously in the discount factor. Yet there are simple absorbing games in which no ε-equilibrium can be selected continuously in the discount factor. The dividing line is convexity: sets of ε-optimal strategies and of mediated (e.g., sunspot) ε-equilibria are convex, while sets of Nash ε-equilibria are not.